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  • A rectangular conducting loop consists of two wires on two opposite sides of length l joined together by rods of length d. The wires each are of the same material but with cross-sections differing by a factor of 2.

A rectangular conducting loop consists of two wires on two opposite sides of length l joined together by rods of length d. The wires each are of the same material but with cross-sections differing by a factor of 2. The thicker wire has a resistance R and the rods are of low resistance, which in turn are connected to a constant voltage source V_{0}. The loop is placed in a uniform magnetic field B at45^{\circ} to its plane. Find T, the torque exerted by the magnetic field on the loop about an axis through the centres of rods.

Answers (1)

To find the net torque it is necessary to calculate the magnetic forces and torques after analysis of the direction of current in both wires.

According to the problem, the thicker wire has the resistance R, then the other wire has resistance 2Ras the wires have resistance 2R as the wires of the same material so their resistivity remains the same.

Now, the force and hence, torque on the first wire is given by 

F_{1}=i_{1}lB\sin 90^{\circ}=\frac{V_{0}}{2R}IB

\tau _{1}=\frac{d}{2\sqrt{2}} F_{1} =\frac{V_{0}IdB}{2\sqrt{2}R}

Similarly, the force hence torque on the other wire is given by

F_{2}=i_{2}lB\sin 90^{\circ}=\frac{V_{0}}{2R}IB

\tau _{2}=\frac{d}{2\sqrt{2}} F_{2} =\frac{V_{0}IdB}{4\sqrt{2}R}

So, net torque \tau =\tau_{1}-\tau_{2}

\tau =\frac{l}{4\sqrt{2}}\frac{V_{0}AB}{R}

Where A is the area of the rectangular coil.

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