Question

Figure 29-27 shows cross-sections of two long straight wires; the left-hand wire carries current i1 directly out of the page. If the net magnetic field due to the two currents is to be zero at point P, (a) should the direction of current i2 in the right-hand wire be directly into or out of the page, and (b) should i2 be greater than, less than, or equal to i1?

Short answer

(a) The direction of current i₂ in the right-hand wire is directly into the page.

(b)i₂ should be greater than i_1.

Step-by-step solution

Given

• Figure 29-27
• Current i₁ is directing out of the page.
• Net magnetic field at point P is zero.

Determining the concept

Find the direction of magnetic fields corresponding to currents. Then applying the given condition, find the direction ofi_2.Then, using the formula for magnetic field at a point due to a current in the wire, find whether i₂ is greater than, less than, or equal toi_1.

Right Hand Rule states that if we arrange our thumb, forefinger and middle finger of the right-hand perpendicular to each other, then the thumb points towards the direction of the motion of the conductor relative to the magnetic field, the forefinger points towards the direction of the magnetic field and the middle finger points towards the direction of the induced current.

Formulae are as follow:

$B=\frac{{\mu }_{0}I}{2\pi r}$

Where, B is magnetic field, r is radius, Iis current, 𝛍is permeability.

(a) Determining should the direction of current i2 in the right-hand wire be directly into or out of page

According to the right-hand rule, if i₁is directed out of the page, the corresponding magnetic field is in the downward direction.

It is given that the net magnetic field at point P is zero. This field is due to i₁and i₂.

To satisfy this condition, the magnetic field corresponding to i₂ in the right-hand wire should direct into the page.

Hence, the direction of current i₂in the right-hand wire is directly into the page.

(b) Determining should the i2 be greater than, less than, or equal to i1

The magnitude of the magnetic field due to both currents should be equal.

But the distance ofi₂from point P is greater than that of i₁.

The magnetic field at point r due to currentI in the wire is given by,

$B=\frac{{\mu }_{0}I}{2\pi r}$

This implies that the magnetic field is inversely proportional to the distance of a point from the wire and directly proportional to the current.

Hence,i₂ should be greater than i_1.

Therefore, the direction of the current can be found using the right-hand rule. The magnetic field depends on the current of the wire and the distance of a point from the wire.