Calculating the Magnetic Field Due to a Current — Question 4

Question 4

A field null between unequal parallel currents locates their current ratio

Two infinitely long wires parallel to +z cross the xy plane at x=0 and x=d. Their positive currents are I1 and I2. A field null is measured at x=alpha d on the segment between them, with $0<\alpha<1$. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Find I2/I1 and identify the directions of the two fields at the null.

2. If alpha=.300 and I1=1.50 A, find I2. Locate the null if I2 alone is reversed.

Question 4 - Solution

A field null between unequal parallel currents locates their current ratio

(a) Method and calculation

$$\vec B_1=\frac{\mu_0 I_1}{2\pi x}\hat y,\quad\vec B_2=-\frac{\mu_0 I_2}{2\pi(d-x)}\hat y.$$$$\frac{I_1}{\alpha d}=\frac{I_2}{(1-\alpha)d}\Rightarrow\boxed{I_2/I_1=(1-\alpha)/\alpha}.$$

(b) Interpretation and further calculation

$$I_2=1.50(0.700/0.300)=\boxed{3.50\,\mathrm A}.$$$$\frac{I_1}{x}-\frac{I_2}{x-d}=0\Rightarrow x=\frac{I_1d}{I_1-I_2}=\boxed{-.750d}.$$After reversal, the null lies outside, on the side of the weaker current.

Physical check

Between opposite currents the fields reinforce, so the null cannot remain between the wires.