Question

A conductor is placed along z-axis carrying current in z direction in uniform magnetic field directed along y-axis. The magnetic force acting on the conductor is directed along:

• A. positive x-axis
• B. positive y-axis
• C. positive z-axis
• D. negative x-axis

Hint:

Use Fleming's Left-Hand Rule as a quick check. Point your Forefinger in the direction of the Magnetic Field (y-axis). Point your Middle finger in the direction of the Current (z-axis). Your Thumb will point in the direction of the Force, which will be the negative x-axis.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem involves finding the direction of the magnetic force on a current-carrying conductor placed in a uniform magnetic field. The direction of this force is determined by the vector cross product of the current direction and the magnetic field direction, often visualized using the right-hand rule.

Step 2: Key Formula or Approach:
The magnetic force \(\vec{F}\) on a straight conductor of length \(\vec{L}\) carrying current \(I\) in a uniform magnetic field \(\vec{B}\) is given by: \[ \vec{F} = I (\vec{L} \times \vec{B}) \] The direction of the force is given by the direction of the cross product \(\vec{L} \times \vec{B}\). We can use the Cartesian unit vectors (\(\hat{i}, \hat{j}, \hat{k}\) for x, y, z axes respectively) to determine this direction.

Step 3: Detailed Explanation:
Given directions:
The current is in the z-direction. So, the direction of the length vector \(\vec{L}\) is along the z-axis, which can be represented by the unit vector \(\hat{k}\).
The magnetic field is in the y-direction. So, the direction of \(\vec{B}\) is along the y-axis, represented by the unit vector \(\hat{j}\).
Calculation of Direction:
The direction of the force \(\vec{F}\) is determined by the cross product \(\hat{k} \times \hat{j}\).
Using the cyclic property of the cross product of unit vectors: \(\hat{i} \times \hat{j} = \hat{k}\)
\(\hat{j} \times \hat{k} = \hat{i}\)
\(\hat{k} \times \hat{i} = \hat{j}\)
And the anti-cyclic property: \(\hat{j} \times \hat{i} = -\hat{k}\)
\(\hat{k} \times \hat{j} = -\hat{i}\)
\(\hat{i} \times \hat{k} = -\hat{j}\)
From this, we find that \(\hat{k} \times \hat{j} = -\hat{i}\).
The vector \(-\hat{i}\) represents the direction along the negative x-axis.

Step 4: Final Answer:
The magnetic force acting on the conductor is directed along the negative x-axis.