Example 01:

A Rowland ring (toroid) of ferromagnetic material of a mean radius 15 cm has 3000 turns of wire wound it. The relative permeability of the material is 1000. What is the magnetic field in a core when a current of 2 A passes through the windings?

Given: mean radius = 15 cm = 0.15 m, Number of turns = 3000, the relative permeability = μ_r = 1000, current through coil = i = 2 A, μ_o = 4π x 10^-7 Wb/Am.

To Find: Magnetic field = B =?

Solution:

n = N/2πr = 3000 / (2π x 0.15)

∴ B = μ_r μ₀ n i = 1000 x 4π x 10^-7 x (3000 / (2π x 0.15) x 2

∴ B = 1000 x 2 x 10^-7 x (3000 /0.15) x 2 = 1000 x 2 x 10^-7 x (20000) x 2

∴ B = 8 T

Ans: Magnetic field = 8 T

Example 02:

A toroidal ring (toroid) of ferromagnetic material of mean radius 15 cm has 3500 turns of wire wound it. The relative permeability of the material is 800. What is the magnetic field in a core when a current of 1.2 A passes through the windings?

Given: mean radius = 15 cm = 0.15 m, Number of turns = 3500, the relative permeability = μ_r = 800, current through coil = i = 1.2 A, μ_o = 4π x 10^-7 Wb/Am.

To Find: Magnetic field = B =?

Solution:

n = N/2πr = 3500 / (2π x 0.15)

∴ B = μ_r μ₀ n i = 800 x 4π x 10^-7 x (3500 / (2π x 0.15) x 1.2

∴ B = 800 x 2 x 10^-7 x (3500 /0.15) x 1.2 = 4.48 T

Ans: Magnetic field = 4.48 T

Example 03:

A Rowland ring of ferromagnetic material has 3000 turns. The inner and outer radii of ring are 11 cm and 12 cm respectively. If a current of 0.7 A is passed through its coils, the magnetic field produced in the core is 2.5 T. What is relative permeability of the core?

Given: mean radius = (11 +12)/2 = 11.5 cm = 0.115 m, Number of turns = 3000, Magnetic field = B = 2.5 T, current through coil = i = 0.7 A, μ_o = 4π x 10^-7 Wb/Am.

To Find:  relative permeability = μ_r =?

Solution:

n = N/2πr = 3000 / (2π x 0.115)

∴ B = μ_r μ₀ n i

∴ 2.5   = μ_r x 4π x 10^-7 x (3000 / (2π x 0.115)) x 0.7

∴ 2.5   = μ_r x 2x 10^-7 x (3000 /0.115) x 0.7

∴ 2.5   = μ_r x 3.652 x 10^-3

∴ μ_r = 2.5/(3.652 x 10^-3) = 6.846 x 10² = 684.6

Ans: The relative permeability = 684.6

Example 04:

A soft iron ring of cross-sectional diameter 8 cm and mean circumference 200 cm has 400 turns of wire wound uniformly around it. Calculate the current necessary to produce magnetic flux of 5 x 10^-4 Wb. Take relative permeability of iron 1800.

Given: Cross-sectional diameter = 8 cm, cross-sectional radius = R = 4 cm = 0.04 m, mean circumference = 2πr = 200 cm = 2 m, Number of turns = 400, Magnetic flux = Φ = 5 x 10^-4 Wb, relative permeability = μ_r = 1800, μ_o = 4π x 10^-7 Wb/Am.

To Find:  current through coil = i =?

Solution:

n = N/2πr = 400 /2 = 200

B = Φ/A = Φ /πR² = (5 x 10^-4) / (3.142 x (0.04)²)

∴ B = ( 5 x 10^-4)/(3.142 x 16 x 10^-4) = 0.09946 T

B = μ_r μ₀ n i

∴ 0.09946   = 1800 x 4π x 10^-7 x 200 x i

∴ 0.09946   = 1800 x 4 x 3.142 x 10^-7 x 200 x i

∴ 0.07812   = 0. 4524 i

∴ i = 0.099462/0. 4524 = 0.22 A

Ans: The current through coil is 0.22 A

Example 05:

A soft iron ring of cross-sectional area 1.5 cm² and mean circumference 30 cm has 240 turns of wire wound uniformly around it. Calculate the relative magnetic permeability if the current necessary to produce magnetic flux of 7.5 x 10^-4 Wb is 2 A.

Given: Cross-sectional area = 1.5 cm² = 1.5 x 10^-4 m², mean circumference = 2πr = 30 cm = 0.3 m, Number of turns = 240, Magnetic flux = Φ = 7.5 x 10^-4 Wb, current through coil = i = 2 A, μ_o = 4π x 10^-7 Wb/Am.

To Find:   relative permeability = μ_r =?

Solution:

n = N/2πr = 240 /0.3= 800

B = Φ/A = (7.5 x 10^-4) / (1.5 x 10^-4) = 5 T

B = μ_r μ₀ n i

∴ 5   = μ_r x 4π x 10^-7 x 800 x 2

∴ 5   = μ_r x 4 x 3.142 x 10^-7 x 800 x 2

∴ 5   = μ_r x 2.01 x 10^-3

∴ μ_r = 5/( 2.01 x 10^-3) = 2488

Ans:  relative permeability is 2488.

Example 06:

The radius of an Amperian loop in a toroid of 2000 turns is 10 cm. If the magnetic induction inside the toroidal space is 0.08 T. What is the magnitude of the current flowing?

Given: mean radius = 10 cm = 0.1 m, Number of turns = 2000, Magnetic field = B = 0.08 T, μ_o = 4π x 10^-7 Wb/Am.

To Find:  current through coil = i =?

Solution:

n = N/2πr = 2000 / (2π x 0.1)

∴ B = μ_r μ₀ n i

∴ 0.08   = 1 x 4π x 10^-7 x (2000 / (2π x 0.1)) x i

∴ 0.08   = 2 x 10^-7 x (20000) x i

∴ 0.08   = 4 x 10^-3 i

∴ i = 0.08 / (4 x 10^-3) = 20 A

Ans: The current through coil is 20 A

Example 07:

A toroid has 5000 turns wound upoin it and carries a current of 15 A. What is the magnetic flux density inside the torroid at point 25 cm from the centre of toroidal circle?

Given: distance from centre = r = 25 cm = 0.25 m, Number of turns = 5000, current through coil = i = 15 A, μ_o = 4π x 10^-7 Wb/Am.

To Find:   Magnetic field = B =?

Solution:

∴ B = μ_r μ₀ n i

∴ B = μ_r μ₀ (N/2πr) i

∴ B   = 1 x 4π x 10^-7 x (5000 / (2π x 0.25)) x 15

∴ B   = 2 x 10^-7 x (5000 /0.25) x 15

∴ B   = 2 x 10^-7 x 20000 x 15

∴ B   = 0.06 T

Ans: Magnetic flux density is 0.06 T

Example – 08:

A closed wound narrow toroid has 500 turns wound upoin it and carries a current of 5 A. What is the magnetic flux density inside the torroid at point 5 cm from the centre of toroidal coil?

Given: distance from centre = r = 5 cm = 0.05 m, Number of turns = 500, current through coil = i = 5 A, μ_o = 4π x 10^-7 Wb/Am.

To Find:   Magnetic field = B =?

Solution:

∴ B = μ_r μ₀ n i

∴ B = μ_r μ₀ (N/2πr) i

∴ B   = 1 x 4π x 10^-7 x (500 / (2π x 0.05)) x 5

∴ B   = 2 x 10^-7 x (500/0.05) x 5

∴ B   = 2 x 10^-7 x 10000 x 5

∴ B   = 0.1 T

Ans: Magnetic flux density is 0.1 T

Example 09:

A toroid is wound on a paramagnetic substance of susceptibility 3 x 10^-4. What will be the percentage increase in magnetic field of toroid?

Given: susceptibility = χ =3 x 10^-4,

To Find:   % Change in magnetic field =?

Solution:

We have μ_r = χ + 1= 3 x 10^-4 + 1 = 0.0003 + 1 = 1.0003

Ans: Percentage increase in the magnetic field is 0.03 %

Example – 10:

A toroid is wound on a paramagnetic substance of susceptibility 6.8 x 10^-5. What will be the percentage increase in the magnetic field of the toroid?

Given: susceptibility = χ = 6.8 x 10^-5,

To Find:   % Change in the magnetic field =?

Solution:

We have μ_r = χ + 1= 6.8 x 10^-4 + 1 = 0.000068 + 1 = 1.000068

Ans: Percentage increase in the magnetic field is 6.8 x10^-3 %

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Science > Physics > Magnetic Effect of Electric Current >Numerical Problems on Toroids or Rowland Ring

The post Numerical Problems on Current Through Toroids or Rowland Ring appeared first on The Fact Factor.

In this article, we shall study Biot-Savart’s Law and its application to find the magnetic induction at a point due to current carrying infinitely long conductor and the magnetic induction at the centre of a current-carrying circular coil.

Magnetic Induction:

The magnetic induction at any point in the magnetic field is defined as the magnetic flux passing through the unit area at that point. It is denoted by letter “B”. It is a vector quantity. Its S.I. unit is Wb/m² or tesla (T).

Mathematically, B = ∅ /A

Where B = Magnetic induction, ∅= Magnetic flux

A = Area through which magnetic flux is passing

Magnetic Induction Due to Current-Carrying Conductor:

Biot Savart’s law (Laplace’s Law):

The magnetic induction at a point near current-carrying conductor is directly proportional to (a) the current in the element (I), (b) the length of the element (dl), (c) the sine of the angle between the element and the straight line joining the element to the point, and inversely proportional to the square of the distance between the element and the point.

Explanation:

Consider a conductor of any shape in the form of wire. Let ‘i’ be the current through the conducting wire. Let ‘dl’ be a small element of the conductor, then the quantity i. dl is known as the current element of the conductor. Let P be the point at which magnetic induction due to current carrying conductor is to be found. Let be the position vector of point  P with respect to the current element i.dl. Let θ be the angle between dl and r, then by Biot Savart’s law the magnetic induction due to the small element dl is given by

dB ∝ i           …………….. (1)

dB ∝ dl               ……………(2)

dB ∝ sinθ        ……………(3)

dB ∝ 1/r²            ……………(4)

From equations (1), (2), (3) and (4)

This is a mathematical expression for the Biot-Savart’s law.

This is a mathematical expression for the Biot-Savart’s law in the vector form.

The total magnetic induction due to current in the whole conductor is given by

Magnetic Induction at a Point Near an Infinitely Long Straight Current-Carrying Conductor:

Let us consider infinitely long straight conductor carrying current i. Let P be the point at which magnetic induction due to the current-carrying conductor is to be found. Let us consider a small element dl of the conductor at O. Let vector AP or vector r be the position vector of point  P with respect to the current element. Let θ be the angle between the position vector of the point P and the current element. Let us draw PC perpendicular to the length of the conductor is drawn such that PC = R  and OC = l.  Now, angle COP = π – θ

Let dB be the magnetic induction at point ‘P’ due to the current-carrying conductor it is given by Biot Savart’s law.

This is an expression for magnetic induction at the point near the infinitely long straight conductor.

Magnetic Induction at Centre of Current-Carrying Circular Coil:

Consider a current-carrying coil having radius R and having a single turn. Let i be the current through this circular coil.  Let O be the centre of the circular coil at which magnetic induction is to be found.  Let dl be the small element of the current-carrying coil at P, let θ be the angle between the current element and line joining the current element with point O.  In this case, θ = 90⁰.

Magnetic induction at point O due to current element i. dl is given by Biot-Savart’s law

The total magnetic induction at “O” can be found by integrating both sides.

This is an expression for magnetic induction at the centre of the current-carrying circular coil.

Previous Topic: Magnetic Effect of Electric Current

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Science > Physics > Magnetic Effect of Electric Current > Biot-Savart’s Law

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In this article, we shall study the magnetic effect of electric current and its applications.

Oersted’s Experiment:

Oersted performed an experiment to show the magnetic effect of electric current. He took long conducting wire and aligned it in magnetic meridian i.e. along the north-south direction. He put the magnetic needle exactly below the conductor. The magnetic needle aligned itself in the north-south direction such that both the conductors and the magnetic needle are parallel to each other.

When a current is passed through the conductor there was a deflection in the magnetic needle. When the current was switched off the magnetic needle came back to its original position. It shows that the magnetic field produced due to the flow of electric current. When the direction of the current has reversed the direction of the deflection of the magnetic needle also reversed. It shows that the direction of the magnetic field produced by the current in the conductor gets reversed by reversing the direction of the current through the conductor.

When the magnetic needle is put exactly above the conductor keeping the direction of the current same, the direction of the deflection of the magnetic needle was reversed.

When the distance of the magnetic needle is increased the deflection of the magnetic needle decreased. Thus the strength of a magnetic field at a point created by the current-carrying conductor depends on the distance of the point from the conductor.

Conclusions:

• Initially, the magnetic needle aligns itself in the magnetic meridian due to the action of the earth’s magnetic field on the needle. After passing a current through the conductor there is a deflection in the magnetic needle and when the current is switched off there is no deflection in the magnetic needle which indicates that due to the flow of current in conductor there is a creation of magnetic field around the conductor.
• If the direction of current is reversed deflection of the needle is also reversed which indicates that if the direction of current in the conductor is reversed there is a reversal of direction of the magnetic field around the conductor.
• The direction of the deflection of the needle depends upon the position of the needle i.e. whether it is kept above or below the conductor.
• If the conductor is aligned in the east-west direction and current is passed through it, then there is no deflection.

To Find the Direction of Deflection of Magnetic Needle:

• Swim rule: Imagine a person swimming along the wire in the direction of current with his face towards the needle, then the north pole of the needle will be deflected towards his left hand.
• Right-hand palm rule:
• Hold your right-hand palm along a conductor such that fingers are indicating the direction of current and palm facing conductor then the north pole deflects in the direction of the outstretched thumb.

Nature of Magnetic Field Due to Current in Conductor:

Plane cardboard was taken and a small hole is made at the centre. The conductor carrying current is passed through this hole. The cardboard is kept horizontal and some iron filings are spread on it. When current is passed through the conductor iron filings get adjusted in the form of concentric circles.

It means the magnetic field produced by the current-carrying conductor is in the form of concentric circles.

The Direction of Magnetic Field Due to Current in Conductor:

Right-Hand Thumb Rule or Right-Hand Grip Rule:

Hold the current-carrying conductor in your right hand such that the outstretched thumb indicates the direction of the current, then the direction of curled fingers indicate the direction of the magnetic field created by the conductor.

Right-hand Screw Rule:

Imagine that right-hand screw is held with its axis parallel to the direction of the conductor is rotated with fingers such that the tip of the screw move in the direction of the current. Then the direction in which the fingers rotate the head gives the direction of magnetic induction.

Factors Influencing the Strength of Magnetic Field:

• The magnetic field created due to current carrying straight conductor is in the form concentric circles whose direction is given by Right-hand thumb rule.
• The strength of the magnetic field depends on the current through the conductor. If the current increases, the strength of magnetic fields increases. When the current through the conductor decreases there is a decrease in the strength of the magnetic field.
• The strength of the magnetic field at a point depends on the distance of the point from the current-carrying conductor. If the distance increases the strength of the magnetic field decreases and as the distance decreases the strength of the magnetic field increases.

Electromagnets:

An electromagnet consists of a soft iron piece on which an insulated copper wire is wound. Thus it is a current-carrying coil wrapped around a piece of iron called core.

When an electric current is passed through the coil the arrangement behaves like a magnet. It is a temporary magnet because when the current through the coil is switched off, the arrangement ceases to be a magnet. The strength of electromagnets depends on the number of turns of the coil and the strength of the electric current through the coil. The strength of electromagnets increases with an increase in the number of turns of the coil and also increases with an increase in the amount of the current passed through the coil. By reversing the direction of current the position of poles of electromagnets can be altered.

The end of the coil where the direction of current anticlockwise becomes the North Pole and the end where the direction of the current clockwise becomes the South Pole. Symbolically it is shown as follows.

There are two types of electromagnets viz. a) bar magnet (The iron piece is in the form of a bar) and b) U-shaped magnet (The iron piece is in U-shaped).

Applications of Magnetic Effect of ELectric Current:

Electrical appliances such as the electric doorbell, electric fan, electric motors work on the principle of electromagnets.

• They are used in lifting heavy iron loads and iron scrap.

• They are used to remove iron particles from the wound.
• They are used in the concentration of ores using electromagnetic separation.

• They are used in the preparation of strong permanent magnets used in speakers.

• They are used in loading iron in the furnace.
• Some medical ailments can be detected and cured by electromagnets.

• The principle of electromagnetism is used in high-speed Maglev trains

Force Experienced by Current-Carrying Conductor in Magnetic Field:

The conductor carrying current when subjected to a magnetic field experiences a force. If the conductor is perpendicular to the magnetic field then the direction of the experienced force is perpendicular to both the conductor and the magnetic field. When the conductor is parallel to the magnetic field, it experiences no force.

Solenoids:

A solenoid consists of an insulated copper wire wound around a cylindrical cardboard or a soft iron core. It is similar to the electromagnet.

Electric Bell

The circuit arrangement of an electric bell is as follows

When the Push switch is pressed circuit gets completed. The current flows through the coil. It creates a U-shaped electromagnet which attracts the armature made up of magnetic material towards it. A hammer is attached to the armature. When armature moves towards the electromagnet, the hammer strikes on the gong and the bell rings. But when the armature is moving towards the electromagnet the contact with the adjustment screw breaks. Thus the current in the electromagnet stops. The iron core of electromagnet loses its magnetic properties and armature is thrown back away from the magnet due to the spring to its normal position where there is a contact again with the adjustment screw. Thus the circuit is completed. This process repeats continuously by make and break and the hammer continues to strike the gong till the push button is pressed.

Next Topic: Biot-Savart’s Law and its Applications

Science > Physics > Magnetic Effect of Electric Current > Magnetic Effect of Electric Current

The post Introduction to Magnetic Effect of Electric Current appeared first on The Fact Factor.

The device which uses the electric or magnetic field to guide and accelerate a beam of charged particles to high speed is called a particle accelerator. Charged particles used may be protons or electrons. These high-velocity particles are used in nuclear physics and high energy physics. Depending upon the direction of motion of charged particles, they have classified into two types a) Linear accelerator and b) Circular accelerator or Cyclotron.

In linear accelerator charged particles move in a straight line. To achieve velocity in kilometres, the length of linear accelerators is also in kilometres, thus the length of linear accelerators is very large. In Cyclotron the same distance as in linear accelerators is covered in concentric circles. Hence the size of the cyclotron is greatly reduced. The first cyclotron was developed by Lawrence and Livingston in 1931 at the University of California.

Cyclotron:

Principle:

An electric field can accelerate a charged particle and the magnetic field can throw the particle in a circular orbit such that its frequency of the revolution does not depend on its speed. Thus the radius of circular path increases, the speed of the particles goes on increasing.

Diagram:

Construction:

It consists of a pair of hollow metal chambers labelled D₁ and D₂ which are flat and semicircular in the shape of the letter ‘D’ back to back. Hence these chambers are called ‘Dees’. A source of a charged particle is located at the midpoint of the gap between the two dees.

The dees are connected to the radio frequency oscillator to apply high frequency between the dees. Dees act as electrodes and the potential between the two dees is made to alter very rapidly. The dees are then fixed in a large metal box which is evacuated. The whole apparatus is put between a strong electromagnet which makes the particle to move in a circular path.

Working:

Let us assume a positive charge particle (proton) having a charge ‘q’ is emitted from the source. Let us assume at this instant D₁ is negative. It is accelerated towards D₁. Let v₁ be the velocity of the particle when it reaches D₁. Due to the magnetic field of induction, it starts moving in a circular path in D₁. Let r₁ be the radius of the circular path and ‘m’ be the mass of the charged particles.

After completing the semicircular path, the particle reaches gap again. At this instant, D₂ is made negative.  Let v₂ be the velocity of the particle when it reaches D₂. Due to the magnetic field of induction, it starts moving in a circular path in D₂. Let r₂ be the radius of the circular path.

Dividing equation (1) by (2) we get

Thus the angular velocity and hence the period is constant or it is independent of the velocity of the particle. Thus at every crossing of the gap the velocity of the charged particle increases. It traces a flat spiral of increasing radius. Then at the periphery point, it is deflected on target with very high velocity using a deflecting plate.

Uses:

• The fast-moving charged particles obtained using cyclotron can be used in artificial transmutation.
• The fast-moving charged particles obtained using cyclotron can be further used to obtain high energy fast-moving particles like neutrons, which can be used in nuclear fission.
• The fast-moving charged particles obtained using cyclotron can be used in inducing artificial radioactivity.

Previous Topic: Tangent Galvanometer

Science > Physics > Magnetic Effect of Electric Current > Cyclotron

The post Cyclotron appeared first on The Fact Factor.

In this article, we shall study, the principle, construction, working, sensitivity, and accuracy of the tangent galvanometer.

Principle:

The tangent galvanometer works on the principle of tangent law. The magnetic needle is subjected to two magnetic fields which are perpendicular to each other. One field is due to the horizontal component of the earth’s magnetic field and the other is created by passing the current through the coil of the tangent galvanometer. Under the action of two magnetic fields the needle comes to rest making angle θ with B_H, such that B =  B_H tan θ

Diagram:

Construction:

The tangent galvanometer consists of the following parts.

A coil having a large number of turns:

The coil consists of an insulated copper wire wound on a vertical circular frame. The frame is made up of nonmagnetic material like ebonite.

The frame is fixed vertically on a horizontal base with leveling screws. The coil is provided with two or more coils of a different number of turns. Thus the number of turns can be changed.

A small magnetic needle:

A magnetometer box with a magnetic needle is kept at the centre of the coil in a horizontal plane. The magnetic needle is pivoted at the centre of the coil and can rotate in a horizontal plane.

A light aluminium pointer is pivoted at a right angle to the magnetic needle.  When the needle turns through a certain angle the aluminum pointer also turns through the same angle.

The deflection can be read on a circular degree scale. A mirror is placed below the pointer to avoid error due to parallax.

Adjustment:

• All nearby magnets and magnetic materials are removed away from the instrument.
• The instrument is first leveled using spirit level by adjusting leveling screws at the base so that the needle is exactly horizontal and the coil is exactly vertical.
• The coil is then rotated about its vertical axis so that its plane is parallel to the needle.  Thus the coil remains in the magnetic meridian.
• The magnetometer box is rotated so that the pointer reads 0° – 0°.

Working:

When the current is not passed through the coil the magnetic needle is acted upon by only a horizontal component of the earth and thus the magnetic needle remains in the north-south direction. When the current is passed through the coil a magnetic field is created by the coil. Thus the magnetic needle is now acted upon by two magnetic fields which are at a right angle to each other. Hence the needle will turn through an angle say θ.

Let B_H be the horizontal component of the earth’s magnetic induction and B be the magnetic induction at the centre of the coil due to the current in the coil.  The direction of B will be perpendicular to the plane of the coil and hence perpendicular to B_H. By the tangent law

B = B_H tan θ   …………. (1)

The magnetic induction at the centre of the coil due to current in the coil is given by

Where k is a constant, called the reduction factor of the tangent galvanometer.

∴  i   ∝  tan θ

Thus in a tangent galvanometer, the current through the coil is directly proportional to the tangent of the angle of deflection of the needle.  Due to this characteristic of the galvanometer is known as the tangent galvanometer. Knowing k and θ we can calculate the value of the current through the coil.

The Sensitivity of a Tangent Galvanometer:

The sensitivity of the tangent galvanometer is defined as the ratio of the change in deflection of the galvanometer to the current producing this deflection.

Sensitivity = dθ / di

A galvanometer is said to be sensitive if it gives larger deflection for a small current.

Thus the sensitivity of tangent galvanometer can be increased by

• Increasing the number turns (n) of the coil,
• Keeping the value of cos²θ  small, i.e. deflection should be small,
• Decreasing the magnetic induction (B_H).  It can be done by putting auxiliary bar magnet below the magnetic needle such that it opposes the magnetic induction B_H
• Decreasing the radius (r) of the coil.

Limitations in Increasing Sensitivity:

• If the number of turns of the coil is increased the radius of the coil will not be the same for all turns and the magnetic induction at the centre is not uniform.
• The increase in the number of turns increases the resistance of the coil of the galvanometer.
• If the radius of the coil is made small the magnetic field at the centre will be non-uniform.
• If θ is kept small accuracy decreases. Accuracy is maximum when θ = 45°

Accuracy of Tangent Galvanometer:

Let dθ be the small error in measuring the deflection θ so that the corresponding error in the current calculated is di when measuring a current ‘i’. Then

Now it is clear that the accuracy is maximum when the error is minimum i.e. the quantity di/i  is minimum. i.e. the quantity (sin 2θ)  is maximum. i.e.

sin 2θ = 1

∴   2θ = sin^-1(1)

∴   2θ = 90°

∴   θ = 45°

Thus the accuracy of a T.G. is maximum at a deflection of 45°.

The accuracy can be increased by

• Using light and long pointer.
• Increasing the radius of each turn in the coil. Larger is the radius greater will be the accuracy.  Because due to the larger radius the magnetic induction produced near the centre of the needle will be more and nearly uniform.
• Decreasing the number of turns in the coil. For greater accuracy, the number of turns in the coil must be lesser. Because if the number of turns is large there can be the difference in the radii of different turns and hence current cannot be calculated accurately.

Note:

If we study the conditions of accuracy and sensitivity, we can note that they are contradictory to each other thus accuracy and sensitivity cannot be combined in a single instrument.

Distinguishing Between MCG and a TG:

Previous Topic: Ammeters and Voltmeters

Next Topic: Particle Accelerator (Cyclotron)

Science > Physics > Magnetic Effect of Electric Current > Tangent Galvanometer

The post Tangent Galvanometer appeared first on The Fact Factor.

Ammeter:

An ammeter is an electrical measurement device (apparatus) which is used to measure the electric current in the electrical circuit.

Requirements of Good Ammeter:

• The resistance of an ammeter must be as small as possible.  The ideal resistance of an ammeter should be zero.
• The ammeter should have a range sufficient for measuring a given current.
• It should be sufficiently sensitive (i.e. for a small change in current there should be a reasonable change in deflection).
• It should be direct reading.

Necessity of Shunting for an Ammeter:

We know that an ammeter is used for measuring current.  It is to be connected in series in the circuit for this purpose. If the resistance of ammeter is not very low but considerable, then, when it is connected in a circuit the effective resistance of the circuit will considerably increase. Due to the increase in the value of the effective resistance of the circuit the current in the circuit will decrease compared to the original current in the circuit which was to be measured.  Thus the decreased value of the current will be measured by the ammeter. In order to avoid this, the resistance of the ammeter should be very low. i.e. then the ammeter will be able to measure the current more accurately.

The effective resistance of the parallel combination is always smaller than the least value in the parallel combination. Hence a low-value resistance is connected in parallel across the galvanometer. This process is known as shunting.

Uses of a Shunt in an Ammeter:

• Due to the shunt, the effective resistance of the ammeter will be very low.
• Shunt increases the range of measurement of the current by the galvanometer and hence the range of ammeter is increased.
• Shunt protects the galvanometer coil from being damaged due to the excess flow of current.

Conversion of a Moving Coil Galvanometer into an Ammeter:

A sensitive moving coil galvanometer (pivoted type) is taken. To measure given the range of current a low resistance of suitable value is connected in parallel with it.

The scale is calibrated by comparison with a standard instrument.

Expression for the Value of a Shunt in an Ammeter:

Let G be the resistance of the galvanometer coil.  Let I_g be the current required to give full-scale deflection in the galvanometer.  Let S be the resistance to be connected in parallel with the galvanometer coil so as to measure a maximum current I. Let I_s be the current through the shunt.

This is the expression for the low-value resistance which is to be connected in parallel with the galvanometer to convert it into an ammeter to measure the given current.

Voltmeter:

A voltmeter is an electrical measurement device (apparatus) which is used to measure the potential difference in the electrical circuit.

Requirements of Good Voltmeter :

• The resistance of a voltmeter must be as large as possible.  The ideal resistance of an ammeter should be infinity.
• The voltmeter should have a range sufficient for measuring a given potential.
• It should be sufficiently sensitive (i.e. for a small change in voltage there should be a reasonable change in deflection).
• It should be direct reading.

Necessity of Connecting High-Value Resistance in a Voltmeter:

We know that a voltmeter is used to measure the potential difference between two points. (or across a resistance).  It is to be connected in parallel for this purpose. If the resistance of the voltmeter is not very high it will divert a considerable current from the circuit through it, as a result, there will be a fall in potential difference. Instead of measuring the actual potential difference this decreased potential difference will be measured by the voltmeter. In order to avoid this the resistance of the voltmeter should be very high i.e. then the voltmeter will be able to measure the voltage more accurately.

We know that the effective resistance of the series combination is always larger than the largest value in the series combination. Hence a large value resistance connected in series with the galvanometer to convert it into the voltmeter.

Use of a High-Value Resistance in a Voltmeter:

• Due to the high-value resistance, the effective resistance of the voltmeter will be very high.
• High-value resistance increases the range of measurement of the potential difference by the galvanometer and hence the range of voltmeter is increased.
• High-value resistance protects the galvanometer coil from being damaged due to the excess voltage applied.

Conversion of a Moving Coil Galvanometer into a Voltmeter:

A sensitive moving coil galvanometer (pivoted type) is taken. To measure given the range of potential difference a high resistance of suitable value is connected in series with it.

The value of high-value resistance is given by

The scale is calibrated by comparison with a standard instrument.

Expression for Value of a Resistance in a Voltmeter:

Let G be the resistance of the galvanometer coil.  Let I_g be the current required to give full-scale deflection in the galvanometer.  Let R be the resistance to be connected in series with the galvanometer coil so as to measure a maximum potential difference V. Let Is be the current through the shunt.

Diagram:

This is the expression for the high-value resistance which is to be connected in series with galvanometer to convert it into a voltmeter to measure the given potential difference.

Distinguishing Between Ammeter and Voltmeter:

Distinguishing Between Voltmeter and Potentiometer:

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In this article, we shall study principle, construction, working, sensitivity and accuracy of the moving coil galvanometer

Principle:

When a current-carrying coil is suspended in a uniform magnetic field it is acted upon by a torque. Under the action of this torque, the coil rotates and the deflection in the coil in a moving coil galvanometer is directly proportional to the current flowing through the coil.

Construction of Suspended Type Moving Coil Galvanometer:

The suspended type consists of a rectangular coil of thin insulated copper wires having a large number of turns. The coil is suspended between the poles of a powerful horseshoe magnet by a suspension fibre of phosphor-bronze.  A spring is attached to the other end of the coil.  The current enters the coil through the fibre and leaves the coil through the spring. The upper end of the suspension fibre is connected to a rotating screw head so that the plane of the coil can be adjusted in any desired position. The horseshoe magnet has cylindrically concave pole-pieces. Due to this shape, the magnet produces radial magnetic field so that when coil rotates in any position its plane is always parallel to the direction of the magnetic field. When current flows through the coil it gets deflected.

A soft iron cylinder is fixed inside the coil such that the coil can rotate freely between the poles and around the cylinder. Due to the high permeability, the soft iron core increases the strength of the radial magnetic field. A small plane mirror M is fixed to the suspension fibre. This along with lamp and scale arrangement is used to measure the deflection of the coil.

Theory:

Consider a rectangular coil PQRS of single turn having length ‘l’ and breadth ‘b’ suspended in a uniform magnetic field of induction B such that the plane of the coil is parallel to the magnetic field. Let ‘I’ be the current through the coil.

The sides PS and QR being parallel to the magnetic field do not experience any force, but the sides PQ and RS being perpendicular to the magnetic field experience force. The force experienced by each side is given by

F = B I l

By Fleming’s left-hand rule these forces are opposite in direction. As these two forces are equal and opposite they form what is called as a couple and due to which a torque acts on the coil which tries to deflect the coil. The deflection torque is given by,

Torque = Force x Perpendicular distance between the forces.

τ =   F   x   b
∴ τ   =   B I l × b

But l τ b = A, the area of the coil

∴   τ =   B I A

If the coil has ‘n’ turns, then the deflecting torque is given by

∴   τ   = n BIA

Under the action of this torque, the plane of the coil rotates through an angle θ before coming to rest. Due to the radial magnetic field, the plane of the coil is always parallel to the direction of the magnetic field.  Thus at any position, the deflecting torque has constant magnitude. The rotation of the coil produces a twist in the fibre which produces a restoring torque which is directly proportional to the angle of deflection θ.

τ ∝  θ
 ∴    τ  = k  θ

Where k is the torque per unit twist (or torsional constant) of the suspension fibre.

When the coil comes to rest i.e. when it attains equilibrium, the restoring torque will balance the deflecting torque. So in equilibrium position of the coil,

Deflecting torque    =     Restoring torque.
n B I A  = k θ

The quantities in bracket are constant, therefore
∴   I ∝ θ

Thus in a moving coil galvanometer current in the coil is directly proportional to the angle of deflection of the coil.

Advantages of Moving Coil Galvanometers:

• They are not affected by a strong magnetic field.
• They have a high torque to weight ratio.
• They are very accurate and reliable.
• Their scales are uniform.

Disadvantages of Moving Coil Galvanometers:

• The change in temperature causes a change in restoring torque.
• Restoring torque cannot be easily changed.
• There is a possibility of damage to the phosphor bronze fibre or helical restoring spring due to severe stresses.
• Such instruments can only be used for measurement of direct current quantities and can not be used for measurement of alternating current quantities.

Pivoted Type Moving Coil Galvanometer:

Construction :

The rectangular coil of thin insulated copper wires having a large number of turns is pivoted between the poles of a powerful horseshoe magnet. The coil is mounted on a pivot between two supports. The supports are bearings with almost no friction. Two hairsprings are attached one above the coil and other below the coil which controls the rotation of the coil. The two coils are spiralled in opposite directions. Current enters through one coil and leaves through the other. A long pointer is attached to the coil which directly moves over a graduated scale. The whole assembly is fitted in a box with a window through which deflection can be observed.

Note:

The principle, working and theory of pivoted type moving coil galvanometer is the same as suspended type moving coil galvanometer.

Sensitivity of Moving Coil Galvanometer:

The sensitivity of moving coil galvanometer is defined as the ratio of the change in deflection of the galvanometer to the change in the current.

Sensitivity = dθ / di

A galvanometer is said to be sensitive if it gives larger deflection for a small current. The current in moving coil galvanometer is given by

Thus the sensitivity of moving coil galvanometer can be increased by

• Increasing the number turns (n) of the coil,
• Increasing the area (A) of the coil,
• increasing the magnetic induction (B) and
• Decreasing the couple per unit twist (k) of the suspension fibre.

Limitations to Increase in Sensitivity of Moving Coil Galvanometer:

• If the turns of the coil are increased the length of wire and hence the resistance of the coil increases.
• Increasing the area of the coil beyond limit makes the instrument bulky.
• Increase in the number of turns and area of the coil increases the load on suspension fibre. Hence spring higher value of k should be used which decreases the sensitivity of the galvanometer.
• Increasing the strength of magnetic induction leads to an increase in the weight of the apparatus.
• Decreasing the couple per unit twist of the spring leads to a decrease in the strength of the spring.

Accuracy of Moving Coil Galvanometer:

The relative error in the measurement of current is given by di/i. For moving coil galvanometer, the current through it is given by

Thus the error in the measurement of current depends only on the measurement of the deflection in the galvanometer dθ.

For greater accuracy of the galvanometer, the ratio di / i should be small. It is small when the deflection is large. Thus for greater accuracy, the deflection in the galvanometer should be large for small current in it. As the expression of accuracy does not contain the terms n, A, B and k the accuracy is independent of the number of turns of the coil, the area of the coil, the magnetic induction and constant for the spring.

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In this article, we shall study the Ampere’s law and its application in finding magnetic induction due to long straight conductor, solenoid and toroid.

Statement:

The line integral of the magnetic field B around any closed path is equal to μ₀ times the net steady current enclosed by this path.

Mathematically,

Proof:

The magnetic field produced by a long straight conductor is in the form of concentric circles. These circles are in the plane perpendicular to the length of the conductor. The direction of the field is given by right-hand grip rule.

The magnitude of the magnetic field due to a long straight conductor at a point P on the closed-loop of radius ‘a’ is given by

Let us consider a field line through point P and of a very small length ‘dl’ of this field element. Now both B and dl are tangential to the field line at P.

Applications of Ampere’s Law:

Expression for Magnetic Field Due to Solenoid and Toroid:

A cylindrical coil of a large number of turns is called a solenoid. The magnetic field inside the solenoid is given by 

B = μ₀nI.

Where n = number of turns per unit length and

I = Current through the wire of a solenoid

A toroid is a coil which is wound on a torus or a doughnut-shaped structure. For a toroid, 

B = μ₀nI. 

Where n = number of turns per unit circumference and

I = Current through the wire of a toroid

Expression for Magnetic Induction at a Point Due to a Long Straight Conductor Carrying Current:

Let us consider a very long straight conductor carrying electric current ‘I’ as shown. Let us consider a point P at a distance of ‘a’ from the conductor. Let us consider an Amperian loop as an imaginary circle, having radius ‘a’ and passing through point P. Let us consider a very small element of length ‘dl’ of this Amperian loop. Now both  B and dl are tangential to the field line at P.

This is an expression for magnetic induction at a point due to a long straight conductor carrying current.

Expression for Magnetic Induction at a Point  on the Axis of a Long Straight Solenoid and Well Inside it:  

Let us consider a very long straight solenoid having ’n’ turns per unit length and carrying electric current ‘I’ as shown. Let us consider a point well inside the solenoid at which the magnetic induction is to be found.

Consider a rectangular path ABCD of the line of induction such that AB = L = length of the rectangular path. The number of turns enclosed by the rectangle is nL. Hence the total electric current flowing through the rectangular path is nLI. According to  Ampere’s law

Near the ends of the solenoid, the lines of the field are crowded. While for the rest of the space the lines are so widely spaced that the magnetic field is negligible.

This is an expression for magnetic induction at a point on the axis of long straight solenoid and well inside it.

The expression for magnetic induction at a point near and at the end of long straight solenoid is

Expression for Magnetic Induction at a Point  on the Axis of a Toroid:

A toroid is a solenoid bent into a shape of a hollow doughnut. Let us consider a toroidal solenoid of average radius ‘r’ having centre O and carrying current ‘I’. Let us consider an Amperian loop of radius r and traverse it in a clockwise direction. Let N be the number of turns of the toroid. Then the total current flowing through the toroid is ‘NI’. According to Ampere’s law

This is an expression for magnetic induction at a point on the axis of a toroid.

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