A, B, and C are parallel conductors of equal lengths carrying current I, 2I and 3I respectively, as shown in the figure. \( \mathbf{F_1} \) is the force exerted by B on A and \( \mathbf{F_2} \) is the force exerted by C on A. Choose the correct answer
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \vec{F_1} = -1.33\,\vec{F_2} \)
The force per length between two long parallel currents is \( F = \frac{\mu_0 I_a I_b l}{2\pi d} \). Parallel currents attract and antiparallel currents repel. The wires A and B carry currents in the same direction, and A and C in opposite directions (C is downwards).
\( F_1 \) is the force by B on A. B is at distance \( x \), the currents are \( I \) and \( 2I \), and the force is an attraction, so it points from A towards B:
\( \displaystyle F_1 = \frac{\mu_0 (I)(2I)\,l}{2\pi x} = \frac{\mu_0 I^{2} l}{\pi x} \)
\( F_2 \) is the force by C on A. C is at distance \( 2x \), the currents are \( I \) and \( 3I \), and the force is a repulsion, so it points from C towards A, that is, away from C and opposite to \( \vec{F_1} \):
\( \displaystyle F_2 = \frac{\mu_0 (I)(3I)\,l}{2\pi (2x)} = \frac{3}{4}\,\frac{\mu_0 I^{2} l}{\pi x} = \frac{3}{4}F_1 \)
The two forces are opposite in direction, so as vectors
\( \displaystyle \vec{F_2} = -\frac{3}{4}\vec{F_1} \;\Rightarrow\; \vec{F_1} = -\frac{4}{3}\vec{F_2} = -1.33\,\vec{F_2} \)
Note: The options carry vector arrows, so they are read as vectors, and then only Option (4) is correct. Option (3) is true only for the magnitudes (\( F_2 = \tfrac{3}{4}F_1 \)); as vectors it has the wrong sign, because \( \vec{F_1} \) and \( \vec{F_2} \) point in opposite directions.