I am trying but.. I am stuck.. please help..
 

Asked by nitishkrnehu09 | 18th Nov, 2017, 10:45: PM

Expert Answer:

Since the conducting rod moves in magnetic field, EMF is induced across the endpoints of the rod and the induced emf is 
 
E = -vBl
 
due to this induced emf, the current flowing in the loop containing the rod, resistor and connecting wire is (E/R) = (-vBI)/R
here R is resitance 
Since induced current flowing in magnetic field, the force on conductor is B×i×l and this force acts against the movement of conductor
Hence this force retards the movement of the rod and the velocity of the rod as a function of time is worked out as below
 
begin mathsize 12px style F space equals space minus m a space equals space minus m fraction numerator d v over denominator d t end fraction space equals space B cross times i cross times l space equals space B cross times fraction numerator v B h over denominator R end fraction cross times l space
fraction numerator d v over denominator d t end fraction equals space minus space fraction numerator B squared l squared over denominator m R end fraction v

fraction numerator d v over denominator v end fraction equals space minus fraction numerator B squared l squared over denominator m R end fraction d t
B y space i n t e g r a t i n g space v space equals space C space e x p open parentheses negative fraction numerator B squared l squared over denominator m R end fraction t close parentheses
sin c e space v space equals space v subscript 0 space space a t space t equals 0 space semicolon
v space equals space v subscript 0 e x p open parentheses negative fraction numerator B squared l squared over denominator m R end fraction t close parentheses end style
 
Power dissipated in the resistor is calculated as given below
begin mathsize 12px style fraction numerator d w over denominator d t end fraction equals space P space equals space i squared R space equals space fraction numerator v squared B squared l squared over denominator R squared end fraction cross times R space
fraction numerator d w over denominator d t end fraction equals space fraction numerator B squared l squared over denominator R end fraction v subscript 0 superscript 2 space e x p open parentheses negative fraction numerator begin display style 2 B squared l squared end style over denominator begin display style m R end style end fraction t close parentheses

w space equals space integral d w equals space integral subscript 0 superscript infinity fraction numerator begin display style B squared l squared end style over denominator begin display style R end style end fraction v subscript 0 superscript 2 space e x p open parentheses negative fraction numerator begin display style 2 B squared l squared end style over denominator begin display style m R end style end fraction t close parentheses d t space space space equals space 1 half m v subscript 0 superscript 2 space equals space I n i t i a l space K i n e t i c space e n e r g y space o f space t h e space r o d
end style

 
(A) As per above calculation it is true that initial KE will be converted fully into heat energy in Resistor
 
(B) since the induced emf resisting the movement of rod, rod will not continue with velocity v0
 
(C) Due to current in the circuit, there is power dissipation in the resistor. will not be conveted into maganetic energy
 
(D) no workdone for magnetic force, since v×B force is perpendicular to the movement of charges

Answered by  | 11th Dec, 2017, 05:42: PM