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.
Given:
Resistance = R
Constant velocity = v
Formula used:
From previous question, induced emf
w, induced current 
= 
… (i), where E = emf, R = resistance, μ0 = magnetic permeability of vacuum, i = current in the wire, v = velocity of sliding rod, x = distance of centre of rod from wire, l = length of rod.
Now, magnetic force on element 
… (ii), where I = induced current, da = element, B = magnetic field due to infinitely straight wire
… (iii), where μ0 = magnetic permeability of vacuum, i = current in wire, a = distance from wire.
Hence, (ii) becomes
dF = 
integrating with suitable limits, we get
=>Force needed to keep the wire sliding at constant velocity v 
ns)
(b) Current I = E/R = 
here E = emf, R = resistance, μ0 = magnetic permeability of vacuum, i = current in the wire, v = velocity of sliding rod, x = distance of centre of rod from wire, l = length of rod. (Ans)
(c) Rate of heat developed in the resistor = Power(P) = I2R, where I = current, R = resistance
From previous part, I = 
therefore, rate of heat developed = 
(ans)
(d) Power delivered by external agent = rate of heat developed in resistor = 
(ans)