Question

A cylindrical region of radius R contains a uniform magnetic field, parallel to its axis, with magnitude that is changing linearly with time. If r is the radial distance from the cylinder axis, the magnitude of the induced electric field inside the cylindrical region is proportional to:

Answer 1

The induced electric field inside the cylindrical region is directly proportional to r

What is the electric field?

An electric field is the physical field that surrounds electrically charged particles and exerts a force on all other charged particles in the field.

• When a loop of wire is placed in a time-varying magnetic field, an induced electric field arises.

• This induced electric field allows the flow of current in the loop. There is no potential associated with this induced electric field.

Here. a magnetic field with magnitude linearly varies with time.

Radial distance  = r

Area of this region, A = π*r^2

Let the induced electric field be E.

Magnetic flux:

∅(B) = B*A

Let the magnetic field be kt,

where k is some constant and t is time.

(B) = kt * (πr^2)

Using Maxwell's equation of electromagnetism,

$\int\limits^a_b {E} \, dr = -\frac{d∅}{dt}$

$E* 2\pi r = - \frac{d}{dr}(\pi k tr^{2} )$

$E*2\pi r = -\pi kr^{2}$

$E = - \frac{kr}{2}$

Hence

E ∝ r

The induced electric field is directly proportional to r

So, option 2 is correct.

Learn more about Maxwell's equation here

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