Question

Derive an expression for magnetic force \( \vec{F} \) acting on a straight conductor of length \( L \) carrying current \( I \) in an external magnetic field \( \vec{B} \). Is it valid when the conductor is in zig-zag form? Justify.

Hint:

The magnetic force on a current-carrying conductor depends on the angle between the magnetic field and the direction of current. The formula remains applicable even for zig-zag conductors, but the geometry must be considered for accurate calculation.

Answer 1

Step 1: The magnetic force \( \vec{F} \) on a current-carrying conductor of length \( L \) is given by the formula: \[ \vec{F} = I L (\vec{B} \times \hat{l}) \] where: - \( I \) is the current in the conductor, - \( L \) is the length of the conductor, - \( \vec{B} \) is the external magnetic field, - \( \hat{l} \) is the unit vector in the direction of the current. Step 2: The force is calculated by the cross-product of the magnetic field and the direction of current. The magnitude of the force is: \[ F = I L B \sin \theta \] where \( \theta \) is the angle between the magnetic field \( \vec{B} \) and the conductor. Step 3: The direction of the force is given by the right-hand rule, which states that if the right-hand thumb is pointed in the direction of the current, and the fingers are pointed in the direction of the magnetic field, the palm faces in the direction of the force. Step 4: When the conductor is in zig-zag form, the expression still holds, but the total length of the conductor will change. The force calculation will need to account for the effective length and the direction of current in each segment of the conductor. In the zig-zag case, the segments' contribution to the force will depend on the angle between each segment's direction of current and the magnetic field. Thus, while the general formula applies, the overall force will depend on the geometry of the zig-zag shape.