Q57 of 104 Page 306

Figure shows a square frame of wire having a total resistance r placed co-planarly with long, straight wire. The wire carries a current I given by i = i0 sin ωt. Find

(a) the flux of the magnetic field through the square frame,


(b) the emf induced in the frame and


(c) the heat developed in the frame in the time interval 0 to .



Given:


Current in wire i = i0 sin ωt


Length of each side of square loop = a


Distance of one edge from wire = b


Diagram:



Formula used:


Magnetic flux … (i), where = magnetic flux, B = magnetic field, da = area element


Magnetic field due to a long current carrying wire at distance x … (ii), where μ0 = magnetic permeability of vacuum, i = current, x = distance from wire


We consider a strip of width dx at a distance x from the wire.


Now, area element da = a dx, where a = length of loop, dx = width element


Hence, from (i) and (ii),


Flux (ans)


(b) Emf induced in frame where = flux, t = time


From previous part, re,


⇒ = (Ans)


(c)Heat developed in wire, where i = current through frame, r = resistance, t = time


From previous i = E/r =


where E = emf, r = resistance


Hence H =


Now, Given:


Hence, H =


More from this chapter

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55

Figure shows a straight, long wire carrying a current i and a rod of length ℓ coplanar with the wire and perpendicular to it. The rod moves with a constant velocity v in a direction parallel to the wire. The distance of the wire from the centre of the rod is x. Find the motional emf induced in the rod.


 56

Consider a situation similar to that of the previous problem except that the ends of the rod slide on a pair of thick metallic rails laid parallel to the wire. At one end the rails are connected by resistor of resistance R.

(a) What force is needed to keep the rod sliding at a constant speed v?


(b) In this situation what is the current in the resistance R?


(c) Find the rate of heat developed in the resistor.


(d) Find the power delivered by the external agent exerting the force on the rod.


 58

A rectangular metallic loop of length ℓ and width b is placed

coplanarly with a long wire carrying a current i figure. The loop is moved perpendicular to the wire with a speed v in the plane containing the wire and the loop. Calculate the emf induced in the loop when the rear end of the loop is at a distance a from the wire. Solve by using Faraday’s law for the flux through the loop and also by replacing different segments with equivalent batteries.



 59

Figure shows a conducting circular loop of radius a placed in a uniform, perpendicular magnetic field B. A thick metal rod OA is pivoted at the centre O. The other end of the rod touches the loop at A. Tec entre O and a fixed point C on the loop are connected by a wire OC of resistance R. A force is applied at the middle point of the rod OA perpendicularly, so that the rod rotates clockwise at a uniform angular velocity ω. Find the force.