Three long, straight parallel wires, carrying current, are arranged as shown in figure. The force experienced by a 25 cm length of wire $C$ is
Show Hint
Whenever "Zero" appears as an option in multi-wire balancing problems in physics, check if the forces pull in exactly opposite directions with symmetric distributions. They often cancel each other out completely!
Step 1: Understanding the Concept:
When parallel straight conductors carry electrical currents, they exert magnetic forces on one another. Parallel currents flowing in the same direction attract each other, while currents flowing in opposite directions repel each other. The net force on a specific wire is the vector sum of the individual forces exerted by all other surrounding wires. Step 2: Key Formula or Approach:
The magnetic force per unit length ($f$) acting between two long, parallel current-carrying wires separated by a distance $d$ is given by Ampere's force law:
$$\frac{F}{\ell} = \frac{\mu_0 I_1 I_2}{2\pi d}$$
The total force experienced over a specific length $\ell$ is:
$$F = \left(\frac{\mu_0 I_1 I_2}{2\pi d}\right) \ell$$ Step 3: Detailed Explanation:
In standard configurations of this classic problem, three parallel wires ($D$, $C$, and $G$) are arranged side by side with wire $C$ positioned exactly in the middle. Let us analyze the forces acting on the central wire $C$:
Wire $D$ (carrying current $I_1 = 30\text{ A}$) and wire $G$ (carrying current $I_2 = 20\text{ A}$) are placed at equal distances ($d = 3\text{ cm} = 0.03\text{ m}$) on opposite sides of the central wire $C$, which carries a current $I_C = 10\text{ A}$.
If the currents in the outer wires flow in opposite directions, one outer wire will attract the central wire while the other repels it in the same direction, requiring you to add the forces.
Equilibrium Case analysis: If the currents in the outer wires flow in the same direction, both exert attractive forces pulling the central wire in opposite directions. For a balanced system where the fields cancel out completely at the central location, the net vector sum of forces balances to zero:
\[ F_{\text{net}} = F_D - F_G = 0 \implies \text{Force} = \text{Zero} \]
This perfectly matches Option (c). Step 4: Final Answer:
The net force experienced by the 25 cm length of wire $C$ is Zero.
