## Electromagnetic Induction

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JEE Mains

## Use of line integral of electric field

Example: A uniform but time-varying magnetic field exists in a circular region of radius and is directed into the plane of the paper as shown in the figure. Find the magnitude of the induced electric field at point at a distance form the centre of the circular region.

Solution:

The magnitude of the induced electric field inside the solenoid, at a distance from its long central axis is :
A
B
C
D
2

## Circuit rotating perpendicular to the plane of magnetic field

A coil has turns and as it's area. The plane of the coil is placed at right angles to a magnetic induction field of . The coil is rotated through in seconds. The average emf induced in the coil (in milli volts) is:
The induced emf will be given by

mV
A square wire frame of side is placed a distance away from a long straight conductor carrying current . The frame has resistance and self inductance . The frame is rotated by about OO' as shown in figure. Find the electric charge flown through the frame.
A
B
C
D
none of these
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## Induced emf due to varying magnetic field

A magnetic field induction is changing in magnitude in a region at a constant rate . A given mass of copper drawn into a wire and formed into a loop is placed perpendicular to the field. If the values of specific resistance and density of copper are   and respectively, then the induced emf and the resistance in the loop is given by :
cross-sectional area of wire                length of wire

Total resistance of wire
mass

A non-conducting ring having charge q uniformly distributed over its circumference is placed on a rough horizontal surface. A vertical time varying magnetic field is switched on at time . Mass of the ring is 'm' and radius is R. The ring starts rotating after 2 s. The coefficient of friction between the ring and the table is:
A
B
C
D
A uniform but time-varying magnetic field exists in a circular region of radius and is directed into the plane of the paper as shown in the figure. The magnitude of the induced electric field at point at a distance form the centre of the circular region
A
is zero
B
decreases as
C
increases as
D
decreases as
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## Power required to move a conductor in a magnetic field

where B is the magnetic field,
l is the length of the conductor
v is the velocity of the conductor
R is the resistance
Figure shows a square loop of side and resistance . The magnetic field has a magnitude . The work done in pulling the loop out of the field slowly and uniformly in is
A
B
C
D
E
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## Force required to move a conductor in a magnetic field

where B is the magnetic field,r is the overall resistance of the loop.
Example: A rectangular loop with a sliding connector of length m is situated in uniform and constant magnetic field perpendicular to the plane of loop. Resistance of connector is . Two resistances of and are connected as shown in figure. The external force required to keep the connector moving with a constant velocity perpendicular to and in the plane of the loop is :
Since resistances 3 and 6 are in parallel, tnhe equivalent resistance
Motional emf through the rod is where is the magnetic field, is the length of the rod and is the velocity of the rod.
Current through the rod is
Force on the rod
A metallic ring of mass and radius with a uniform metallic spoke of same mass and length is rotated about its axis with angular velocity . in a perpendicular uniform magnetic field of magnitude as shown in figure. The central end of the spoke is connected to the rim of the wheel through a resistor of magnitude as shown. The resistor does not rotate, its one end is always at the center of the ring and other end is always in contact with the ring. A force as shown is needed to maintain constant angular velocity of the spoke then, is equal to (The ring and the spoke has zero resistance)
A
B
C
D
None of these
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## Problem in which circuit is rotating in the plane of magnetic field

Example: In a uniform magnetic field of induction , a wire in the form of semicircle of radius rotates about the diameter of the circle with angular velocity . If the total resistance of the circuit is , then find the mean power generated per period of rotation.

Solution:
Flux

Power:
Mean Power

A square loop of side a is rotating about its diagonal with angular velocity in a perpendicular magnetic field . It has 10 turns. The emf induced is
A
B
C
D
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## Problem based on mutual inductance of different shape

A small square loop of wire of side is placed inside a large square loop of side . If the loops are coplanar and their centres coincide, the mutual induction of the system is directly proportional to

Suppose outer loops carries a current .
field at the center of outer square loop=

therefore,
A circular loop of radius is placed at the centre of current carrying conducting square loop of side . If both loops are coplanar and , then the mutual inductance between the loops will be:
A
B
C
D
A circular wire loop of radius is placed in the x-y plane centered at the origin O. A square loop of side having two turns is placed with its center at along the axis of the circular wire loop, as shown in figure. The plane of the square loop makes an angle of with respect to the z-axis. lf the mutual inductance between the loops is given by , then the value of is
A
B
C
D
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## Energy stored in an inductor due to a magnetic field

Example:
A long wire carries a current . Find the energy stored in the magnetic field inside a volume at a distance from the wire.
Solution:
(energy per unit volume) and

Energy

In the circuit shown (fig), the coil has inductance and resistance. When X is joined to Y, the time constant is during the growth of current. When the steady state is reached, heat is produced in the coil at a rate P. X is now joined to Z. After joining X and Z :
A
the total heat produced in the coil is
B
the total heat produced in the coil is
C
the total heat produced in the coil is
D
the data given are not sufficient to reach a conclusion
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## Behaviour of RL circuits at the time of switching

Example:
A solenoid of inductance with resistance is connected in parallel to a resistance . A battery of emf and of negligible internal resistance is connected across the parallel combination as shown in the figure. At the time , switch is opened, calculate current through the solenoid after the switch is opened.
Solution:
Initially, inductor is fully charged and acts as short circuit. Hence, initially current in the inductor is given by:

When the switch is opened, current across the inductor does not change suddenly.
Hence, current just after opening the switch through the solenoid is given by:

Study the diagram. As soon as the switch is closed, the current through the cell is . After a long time the current through the cell is found to be , then :
A
B
C
D