A conducting rod of length slides with a constant speed on horizontal conducting rails

PHYSICS · Subjective · A Level · Physics · Fields and their consequences · Fields and their consequences

Problem

A conducting rod of length

\ell

slides with a constant speed

v_{0}

on horizontal conducting rails. The rails are in a region where the magnetic field varies along the length of the rails according to

B(x)=B_{0}+kx

, where

x

is the distance of the rod from the left end of the rails (

x=0

t=0

) and

B_{0},\,k

are positive constants. The magnetic field is directed vertically downward. The rails are connected at the left end through a resistor of resistance

R

; the rod has negligible resistance. (a) Derive an expression for the induced emf as a function of time. (b) Obtain the expression for the induced current

I(t)

and state its direction when

k>0

. (c) If

\ell = 0.30\,\text{m}

B_{0}=0.20\,\text{T}

k=0.10\,\text{T m}^{-1}

v_{0}=2.0\,\text{m s}^{-1}

and

R=0.50\,\Omega

, calculate the current when the front end of the rod has reached

x = 1.0\,\text{m}

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