Consider a thin metallic sheet perpendicular to the plane of the paper moving with speed `upsilon` in a uniform magnetic field B going into the plane of the paper (See figure.) If charge den- sities `sigma_1 and sigma_2` are induced on the left and right surfaces, respectively, of the sheet then (ignore fringe effects.)

Options

A. `sigma_(1) = (epsi_(0)vB)/2, sigma_(2) = (-epsi_(0)vB)/2`
B. `sigma_(1) = (-epsi_(0)vB)/2 , sigma_(2) = (epsi_(0)vB)/2 `
C. `sigma_(1) = epsi_(0)vB, sigma_(2) = - epsi_(0)vB`
D. `sigma_(1) = sigma_(2) = epsi_(0)vB`

Answer

c

Explanation

Considering the direction of `vec v xx vec B` we can understand that `sigma_(1)` is positive and `sigma_(2)` is negative. Surface charge density will be of same magnitude on both the faces and let it br `sigma`.
Magnitude of electric field across the width can be written as follows:
`E = sigma/epsi_(0) = Bv rArr sigma = Bv epsi_(0)`
Hence we can write the following:
`simga_(1) = Bvepsi_(0) " and " sigma_(2) = - B vepsi_(0)`
Hence option (c ) is correct