Mathematics

Problem

A science project uses a long, tightly-wound solenoid of 800 turns and length 0.40 m that carries a current of 4 A. A straight conducting rod 5 cm long is placed at the centre of the solenoid so that the rod is perpendicular to the solenoid’s axis and carries a current of 2 A. Answer the following questions related to this arrangement: a. (i) State the formula for the magnetic field inside a long solenoid and name the physical quantities on which it depends. b. (ii) Write the expression for the magnitude of the force acting on a straight conductor of length

L

carrying current

I

when placed in a magnetic field of magnitude

B

at right angles to the field. Explain, in words, how each variable in the two expressions obtained in parts (i) and (ii) influences the magnitude of the force on the rod. c. (iii) Without performing any numerical calculation, predict qualitatively how the magnitude of the force on the rod will change if the solenoid current is doubled while the current in the rod is reduced to half, keeping every other factor unchanged. Give a brief justification for your answer.

Intuitive explanation

The key is recognizing that the rod experiences a force because it carries current through the magnetic field created by the solenoid. First, calculate the solenoid's magnetic field using $B = \mu_0 n I$ , where $n$ is the turn density—this uniform field points along the axis. Then apply the force law $F = BIL\sin\theta$ to the rod: since it's perpendicular to the field, the force depends on both the field strength and the rod's current and length. The direction follows from the right-hand rule.

Intuitive explanation

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