Answer :

Given:


Mass of PQ = m


Resistance of PQ = r


Length of PQ between the two rails = l


Magnetic field = B


Resistance connected to the rails = R


Velocity with which PQ is pushed towards right at t=0 = v0


Formula used:


(a) By Ohm’s law, E = IR’, where E = emf, I = current, R’ = total resistance.


Hence, current … (i)


Now, emf induced due to the moving road in the magnetic field … (ii), where B = magnetic field, l = length of rod, v= velocity of rod


Also, total resistance … (iii), where r = resistance of PQ, R = resistance attached to the rails.


Hence, substituting the values of E and R’ from (ii) and (iii) in (i), we get



Therefore, current in the loop when the speed of the wire PQ is . (Ans)


(b) Now, magnetic force on a current carrying wire … (i), where I = current, l = length of wire, B = magnetic field.


From the previous part, the value of current at an instant when velocity = v is … (ii)


Therefore, from (i) and (ii), magnetic force … (iii)


According to Newton’s second law of motion, … (iv), where F = force, m = mass, a = acceleration.


Hence, equating (iii) and (iv):



Therefore, acceleration of the wire at this instant = (Ans)


(c) Velocity v’ can be expressed as … (i), where v0 = initial velocity, a = acceleration, t = time. We put a negative sign before at since the force is opposite to velocity, and it


Now, from the previous part, we can write acceleration … (ii)


Hence, from (i) and (ii), we can write


But, distance travelled x = vt, where v = velocity, t = time.


Therefore, velocity v as a function of x is (Ans)


(d) We know that,, where a = acceleration,


v = velocity, x = distance, t = time.


Now, from the part (b), acceleration as a function of time




Now, the wire can travel maximum distance when its velocity is v0.


Hence, integrating on both sides, we get



Therefore, maximum distance travelled by the wire


(Ans)


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