Question

problem 20.37
two long straight wires each carry a current i out of the figure toward the viewer (figure 1).
mark the points in which the magnetic field is zero. check all that apply.
figure
1 of 1
i ⨀ 4
1
2 5
3
i ⨀ 6

Explanation:

Step1: Recall magnetic - field formula for a long - straight wire

The magnetic field due to a long - straight wire at a distance $r$ from it is given by $B=\frac{\mu_0I}{2\pi r}$, and the direction of the magnetic field is given by the right - hand rule. For a current coming out of the page, the magnetic field lines are in counter - clockwise circles around the wire.

Step2: Analyze the superposition of magnetic fields

Let the two wires be separated by a distance $d$. At a point between the two wires, the magnetic fields due to the two wires are in opposite directions. The magnetic field due to the left - hand wire at a point $x$ from it and the right - hand wire at a point $d - x$ from it are $B_1=\frac{\mu_0I}{2\pi x}$ and $B_2=\frac{\mu_0I}{2\pi(d - x)}$ respectively. The net magnetic field $B = B_1 - B_2$. When $B = 0$, we have $\frac{\mu_0I}{2\pi x}=\frac{\mu_0I}{2\pi(d - x)}$, which implies $x=\frac{d}{2}$. So, the magnetic field is zero at the mid - point between the two wires. In the given figure, point 2 is the mid - point between the two wires.

Answer:

2