Solution: Magnetic Field and Force on Parallel Conductors
Assumption: Let the distance between conductor (1) and (2) be $d$, and the distance between conductor (2) and (3) be $d$. Thus, the distance between (1) and (3) is $2d$.
Step 1 (Formula / Governing Law & Given Data): [1 Mark]- Magnetic field due to a long straight wire: $B = \frac{\mu_0 I}{2\pi r}$
- Force per unit length between two parallel wires: $f = \frac{\mu_0 I_1 I_2}{2\pi r}$
- Right-Hand Thumb Rule: Used to determine the direction of the magnetic field.
- Fleming's Left-Hand Rule / Force Law: Like currents attract, opposite currents repel.
(i) Net magnetic field at conductor (1):
The magnetic field at conductor (1) is produced by conductors (2) and (3). Assuming currents in (1), (2), and (3) are in the same direction (upwards):
$$B_2 = \frac{\mu_0 (I)}{2\pi d} \text{ (into the plane, } \otimes)$$ $$B_3 = \frac{\mu_0 (3I)}{2\pi (2d)} = \frac{3\mu_0 I}{4\pi d} \text{ (into the plane, } \otimes)$$Since both fields are in the same direction:
$$B_{net} = B_2 + B_3 = \frac{\mu_0 I}{2\pi d} + \frac{3\mu_0 I}{4\pi d} = \frac{2\mu_0 I + 3\mu_0 I}{4\pi d} = \frac{5\mu_0 I}{4\pi d}$$(ii) Net magnetic force per unit length on conductor (1):
Force from (2) on (1): $f_{21} = \frac{\mu_0 (2I)(I)}{2\pi d} = \frac{\mu_0 I^2}{\pi d}$ (Attractive, towards right)
Force from (3) on (1): $f_{31} = \frac{\mu_0 (2I)(3I)}{2\pi (2d)} = \frac{6\mu_0 I^2}{4\pi d} = \frac{1.5\mu_0 I^2}{\pi d}$ (Attractive, towards right)
$$f_{net} = f_{21} + f_{31} = \frac{\mu_0 I^2}{\pi d} + \frac{1.5\mu_0 I^2}{\pi d} = \frac{2.5\mu_0 I^2}{\pi d}$$ Step 3 (Final Concluded Answer): [0.5 Marks]- Net Magnetic Field: $\frac{5\mu_0 I}{4\pi d}$ directed into the plane.
- Net Force per unit length: $\frac{5\mu_0 I^2}{2\pi d}$ directed towards the other conductors (attractive).
💡 EL10 Concept Insight: Think of magnetic fields like invisible circular ripples around a wire; when you have multiple wires, you just add up the strength of these ripples at a specific spot, and since wires with current flowing the same way pull on each other like magnets, we just add their "pulling" forces together.