Step 1: Conceptualization:
This problem requires determining the magnetic force direction on a current-carrying wire within a uniform magnetic field. The force direction is dictated by the vector cross product of the current's direction and the magnetic field's direction, commonly determined using the right-hand rule.
Step 2: Governing Formula:
The magnetic force \(\vec{F}\) on a straight conductor of length \(\vec{L}\) carrying current \(I\) in a uniform magnetic field \(\vec{B}\) is expressed as:\[ \vec{F} = I (\vec{L} \times \vec{B}) \]The force's direction is derived from the cross product \(\vec{L} \times \vec{B}\). Cartesian unit vectors (\(\hat{i}, \hat{j}, \hat{k}\) for the x, y, and z axes, respectively) are utilized to establish this direction.
Step 3: Detailed Analysis:
Input Orientations:
Current flows along the z-axis, thus \(\vec{L}\) aligns with the z-axis, represented by \(\hat{k}\).
Magnetic field is oriented along the y-axis, thus \(\vec{B}\) aligns with the y-axis, represented by \(\hat{j}\).
Direction Calculation:
The force direction \(\vec{F}\) is found via the cross product \(\hat{k} \times \hat{j}\).
Utilizing the cyclic property of unit vector cross products:\(\hat{i} \times \hat{j} = \hat{k}\)
\(\hat{j} \times \hat{k} = \hat{i}\)
\(\hat{k} \times \hat{i} = \hat{j}\)
And the anti-cyclic property:\(\hat{j} \times \hat{i} = -\hat{k}\)
\(\hat{k} \times \hat{j} = -\hat{i}\)
\(\hat{i} \times \hat{k} = -\hat{j}\)
Consequently, \(\hat{k} \times \hat{j} = -\hat{i}\).
The vector \(-\hat{i}\) signifies direction along the negative x-axis.
Step 4: Conclusion:
The magnetic force exerted on the conductor is directed along the negative x-axis.