A conducting rod of length L lies in XY-plane and makes an angle `30^(@)` with X-axis. One end of the rod lies at origin initially. A magnetic field also exists in the region pointing along positive Z-direction. The magnitude of the magnetic field varies with y as `B_(0)((y)/(L))^(3),` where,`B_(0)` is a constant. At some instant the rod stars moving with a velocity `v_(0)` along X-axis
The emf induced in the rod is

Options

A. `(B_(0)v_(0)L)/(64)`
B. `(B_(0)v_(0)L)/(16)`
C. `B_(0)v_(0)L`
D. `64B_(0)v_(0)L`

Answer

A

Explanation

Length along y-axis would be `1 sin 30^(@)=(1)/(2)` length = L
Velocity of rod `= v_(0),` along X-axis variable magnet is field , `B = B_(0)((y)/(L).)^(3),` along Z-axis.
Now, induced emf (formula) (br) E = Blv
where, B,`1_(_|_)` and v are perpendicular to each other,
1 = L / 2
Now, consider a small element of lenght dy at a distance y from origin alaong Y- axis, then emf of element
`dE = B(dy)v_(0)=B_(0)(y^(3))/(L^(3))v_(0).dy`
Emf of the rod,
`E = (B_(0)v_(0))/(L_(0))int_(0)^(L//2) y^(3).dy`
`E = (B_(0)v_(0))/(L^(3))[(y^(4))/(4)]^(L//2)`
`E = (B_(0)v_(0))/(L^(3))[((L//2))/(4)] rArr E = (B_(0)v_(0)L)/(64)`