TOPICS: Electromagnetic Induction: Induced emf, Faraday's law of magnetic induction, Lenz's law, inductance, electric generators, magnetic stored energy, transformers,

Magnetic Flux:

A quantitative measure of the strength of the magnetic field B present in a region, is given by the concept of the magnetic flux. The magnetic flux F through a planar area A is given by

F = BAcosq .....................................

where q is the angle between direction of magnetic field B and normal to the area A. A time varying magnetic flux in a region produces an Induced Electromotive Force (Induced emf) on a conductor placed in it. If a change in the magnetic flux DF occurs in a time Dt, then the induced emf is,

Œ = - DF/Dt ..........................................

The unit of magnetic flux is webers (W). Suppose a coil of N turns of wire is subjected to a time varying magnetic flux, then the induced emf between the two terminals of the coil is,

Œ = -N(DF/Dt) ......................................

The above expression is known as Faraday's law of electromagnetic induction. The negative sign in the expression for induced emf indicates that it always opposes the change in the flux that produced it. If a magnetic flux through a loop of wire is increasing in time, the resulting induced current generated in the loop must be directed so as to oppose the incresing flux. If the flux is decreasing through the loop, the induced current must try to restore the decreasing flux which produced it. This principle is known as Lenz's law.

In the simplest case, we could have a loop of wire (area A) at rest and the magnetic field changing in time. This brings a change in the magnetic flux that generates an induced emf in the loop according to Faraday's law of induction. Hence,

|Œ| = A(DB/Dt)

Here, the magnetic field is directed normal to the plane of the loop.

In a second situation, a straight conductor of length L moving at velocity v in a direction normal to the magnetic field B can generate an induced emf given by,

|Œ| = BLv

Here B, v and length of the conductor are mutually perpendicular to each other. The induced emf produced in this case, is the result of time varying area swept out by the conductor. By Faraday's law of induction, this means a change in the flux.

In a third situation, the loop of wire can be rotated in a fixed magnetic field to bring about a change in the flux. An induced emf will appear in the loop.

TEST YOURSELF:

1. (a) State Faraday law of induction. Show, how any of the three quantities in the magnetic flux can be changed to produce electromotive force.

(b) State Lenz's law.

4. A circular coil of radius 10 cm and 100 turns with a resistance of 100 ohm, is placed in a magnetic field that is normal to the circular area of the coil. At what rate must the magnetic field change with time if an induced current of 100 A is to appear in the coil?

4. A 100 turn coil has a radius of 7.5 cm and a resistance of 50 ohm. At what rate must a perpendicular magnetic field change to produce a current of 5 A in the coil?

2. A solenoid has a length of 0.25m, a radius of 0.01m and 400 turns. It carries a 3 Amp. current. Find,

(a) Flux through the solenoid

(b) The self inductance of the solenoid

(c) The induced EMF in the solenoid if the current changes at 150 Amp./sec.

1. A slide-wire electric power generator is made of a U-shaped wire and a sliding rod of length L as shown. The loop carries a current I. The constant magnetic field B is directed into the plane of the figure. The rod slides over the U-shaped wire with speed v.

(a) Derive an expression of the induced emf produced at the two ends of the rod.

(b) Calculate the force and power needed to move the rod if v = 7 m/s,

B = 1.5 T, L= 2.5 m and I = 2 A.

1. A slide-wire electric power generator is made of a U-shaped wire and a sliding rod of length L as shown. The constant magnetic field B is directed into the plane of the figure. The rod slides over the U-shaped wire with speed v.

(a) Derive an expression for the induced emf at the two ends of the rod.

(b) Calculate the magnitude and direction of the force exerted on the rod.

The resistance in the loop, R = 0.8 ohm, speed v = 41 m/s, B = 1.2 T and L = 0.065 m.

1. A long thin solenoid has 800 turns per meter and a radius of 5.0 cm. The current in the solenoid is increasing at a uniform rate of 50 A/s. Find the induced electric field

(a) inside the solenoid at a radius of 0.5 cm from the axis of the solenoid

(b) outside the solenoid at a radious of 10 cm from the axis.of the solenoid.

1. (a) State Faraday's law of induction. Show clearly, how many different ways can the magnetic flux be changed in time, to produce electromotive force.

(b) State Lenz's law. A rectangular loop of wire is made up of a fixed side of length 0.75 m and a moving metal bar sitting on it. The loop is placed in constant magnetic field of 0.045 T directed into the page of the paper and, perpendicular to the loop area. Find an expression of power required to move the metal bar with speed v = 25 m/s. See figure below.

2. A solenoid has a length of 0.25 m, a radius of 0.01m and 400 turns. It carries a 3 Amp. current. Find,

(a) the flux through the solenoid

(b) the self inductance of the solenoid

(c) the induced EMF in the solenoid if the current changes at 150 Amp./sec.

3. Starting with the Faraday's law of induction, derive an expression which shows time varying magnetic field produces electric field.

4. Derive the expression for self inductance of a solenoid. Calculate the self inductance if the is 0.25 m long, has 2000 turns and has area of cross section of 0.0025m2.

5. A 100 turn coil has a radius of 7.5 cm and a resistance of 50 ohm. At what rate must a perpendicular magnetic field change to produce a current of 5 A in the coil?

6. An electric field is present out of the page in a circular area of radius

R = 0.5 m. If the electric field changes as dE/dt = 25 V/m.s, find the induced magnetic field B at a radius of,

a) 0.75 m

b) 0.25 m.

7. The instantaneous electric field in the electromagnetic waves present in a certain empty space is represented as follows,

E = 15cos (200t - 105) V/m

Write down,

a) the amplitude, frequency, wavelength and period of the wave

b) the phase angle and initial phase (constant)

c) average electromagnetic energy density

d) Poynting vector in this space

2. A closely wound rectangular coil of 50 turns has dimensions of 12cmx25cm. In a time of 0.08 s, the plane of the coil is rotated from a position of 45o with a magnetic field (B = 0.975 T) to a position of 90o with the field. Find the average emf induced in the coil.

3. A horizontal metal rod of length L = 0.5 m moves at a speed of 6 m/s perpendicular to a magnetic field of 0.5 T. Draw a diagram.

(a) Find the induced emf in the rod.

(b) Which end of the rod is at a higher potential?

4. A long thin solenoid has 800 turns per meter and a radius of 2.0 cm. The current in the solenoid is increasing at a uniform rate of 50 A/s. Find the induced electric field near the center of the solenoid at a radius of

(a) 0.4 cm from the axis of the solenoid

(b) 2.5 cm from the axis.of the solenoid.

2. (a) Derive an expression for the self inductance of a rectangular solenoid of length l, area of cross section A and n turns/m.

(b) Calculate the self inductance of a long rectangular solenoid of dimensions 3cmx5cmx50cm and, n = 500 turns per meter.

3. A 100 turn coil has a radius of 7.5 cm and a resistance of 50 ohm. The direction of magnetic field present is normal to the plane of the coil. At what rate must the magnetic field change in magnitude to produce an induced current of 5 A in the coil?

4. A circular coil of radius 10 cm and 100 turns with a resistance of 100 ohm, is placed in a magnetic field of 2.5 T directed normal to the plane of the coil. The coil is rotated at a frequency of 30 Hz. What is the maximum induced current in the loop?

3. Starting with the Faraday's law of induction, derive an expression which shows time varying magnetic field produces electric field.

4. A 100 turn coil has a radius of 7.5 cm and a resistance of 50 ohm. At what rate must a perpendicular magnetic field change to produce a current of 5 A in the coil?

4. A circular coil of radius 10 cm and 100 turns with a resistance of 100 ohm, is placed in a magnetic field that is normal to the circular area of the coil. At what rate must the magnetic field change with time if an induced current of 100 A is to appear in the coil?

6. A long co-axial cable consists of two cylindrical thin hollow conducting shells placed along positive z-axis. The inner shell of radius 0.05m, carries a current 10A along+z-axis. The outer shell of radius 0.2m, carries a current of 5A along negative z-axis. Find the magnetic field at,

(a) 0.01 m from the center of the co-axial cable

(b) 0.08 m from the center of the co-axialcable

(c) 0.25 m from the center of the co-axial cable

7. A solenoid has a length of 0.25m, a radius of 0.01m and 400 turns. It carries a 3 Amp. current. Find,

(a) Flux through the solenoid

(b) The self inductance of the solenoid

(c) The induced EMF in the solenoid if the current changes at 150 Amp./sec.