ABCDA is a closed loop of conducting wire consisting of two semicircular sections, the part ABCD lying in the XY plane and the part CDA lying in the YZ plane, both the parts having the centre at origin O (see the diagram). This loop is placed in a uniform field `vec(B)` which varies with time.
Radius of the semicircular section is 0.5 m

If `vec(B)` be directed along the +z axis and increases at the rate of `10^(-2) ("Tesla")/("second")`. the magnitude and sense of the induced emf in the loop as seen along the field direction will be

Options

A. zero
B. `(5pi)/(4sqrt(2))` mV in the counterclockwise sense
C. `(5 pi)/(4sqrt(2))` mV in the clockwise sense
D. `(5pi)/(2sqrt(2))`mV in the clockwise sense.

Answer

B

Explanation

Since the field is directed along the z axis, there will be no flux through the semicircular section ADC.
The flux through section `ABC=phi=B.A` or `phi=B*(pia^(2))/(2)` direction along z-axis.
The induced emf `e=-(dphi)/(dt)=-(pi a^(2))/(2) |(dvec(B))/(dt)|`
Subsituting value `a=0.5, (dB)/(dt)=+10^(-2)T//s`
`|e|=(pi a^(2))/(2) |(dB)/(dt)|=(pi(0.5)^(2))/(2) 10^(-2)V=(5pi)/(4)mV`
When looking along field directon (i.e) along z-axis this will be counter clockwise sence, since the field increase with time.