Question: Source of force on a current-carrying conductor and force between two parallel conductors.
(a) (i) Source of force: The force on a current-carrying conductor in a magnetic field arises due to the Lorentz force acting on the individual moving charge carriers (electrons) within the conductor, which is then transmitted to the conductor as a whole through collisions with the lattice ions.
Step 1 (Governing Law & Setup): [1 Mark]
Consider two long, straight, parallel conductors $A$ and $B$ carrying currents $I_1$ and $I_2$ separated by a distance $d$. The magnetic field $B_1$ produced by conductor $A$ at the position of conductor $B$ is given by Ampere’s Circuital Law:
$$B_1 = \frac{\mu_0 I_1}{2\pi d}$$Step 2 (Mathematical Working): [3 Marks]
The force $F$ on a segment of length $L$ of conductor $B$ carrying current $I_2$ placed in magnetic field $B_1$ is given by $F = I_2 (\vec{L} \times \vec{B_1})$. Since the current and magnetic field are perpendicular:
$$F = I_2 L B_1 \sin(90^\circ) = I_2 L \left( \frac{\mu_0 I_1}{2\pi d} \right)$$The force per unit length ($f = F/L$) acting on conductor $B$ is:
$$f = \frac{\mu_0 I_1 I_2}{2\pi d}$$By Fleming's Left-Hand Rule, the force is attractive if currents are in the same direction and repulsive if they are in opposite directions.
Step 3 (Definition of Ampere): [1 Mark]
Definition: One ampere is that constant current which, if maintained in each of the two straight parallel conductors of infinite length and negligible circular cross-section, placed one metre apart in vacuum, would produce on each of these conductors a force equal to $2 \times 10^{-7}$ newtons per metre of length.
💡 EL10 Concept Insight: Think of the wires like two magnets; the moving electricity creates a magnetic field around each wire, and these fields push or pull on each other just like the North and South poles of two bar magnets would.