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 \( \vec{F} \) exerted on a current-carrying conductor of length \( L \) is defined by the equation: \[\vec{F} = I L (\vec{B} \times \hat{l})\]Here, \( I \) represents the current in the conductor, \( L \) is the conductor's length, \( \vec{B} \) is the applied magnetic field, and \( \hat{l} \) is the unit vector indicating the current's direction.Step 2: The force is determined by the cross-product of the magnetic field and the current's direction. Its magnitude is calculated as: \[F = I L B \sin \theta\]where \( \theta \) signifies the angle between \( \vec{B} \) and the conductor.Step 3: The right-hand rule dictates the force's direction: orient your right thumb along the current's path and your fingers along the magnetic field; your palm will then point in the force's direction.Step 4: For a zig-zag conductor, the fundamental formula remains applicable, but the total conductor length necessitates adjustment. Force calculation must incorporate the effective length and the directional current within each segment. In a zig-zag configuration, each segment's force contribution is contingent on the angle between that segment's current direction and the magnetic field. Consequently, while the general formula applies, the resultant force is geometry-dependent, influenced by the specific zig-zag shape.