Question

Two long parallel metallic wires with a resistance
'R' form a horizontal plane. A conducting rod AB
is on the wires shown in figure. The space has
magnetic field pointing vertically downwards. The
rod is given an initial velocity 'v'. There is no friction
in the wires and the rod. After a time 't' the velocity
v of the rod will be such that:-
A
R®B
(1) v > v. (2) v < V6 (3) v = v. (4) v =-v.​

Answer 1

Answer:

Explanation:

Solution :

(a) If the rod has instantaneous velocity v at a distance x from R, the include emf is Bvd.

∴ Induced I=BvdR+2λx=constant

∴ Velocity v=(R+2λx)iBd

Megnetic force on the rod =Bid.

This force will be opposite to F. Hence, net force acting in rod,

F−Bid=mdvdt

or F−Bid=mdvdxdxdt=mvddx{(R+2λx)ibd}=mv2λiBd

∴ F=Bid+2mλ(R+2λx)B2d2i2

(b) Work done per second per second =Fv

Heat produced per second =i2(R+2λx)

Requiredratio=i2(R+2λx)BdF(R+2λx)i=iBdF

=BidBid+2mλ(R+2λx)i2B2d2

=11+2mλ(R+2λx)i2B3d3