Chapter 30, Problem 30.1QQ

Consider the magnetic field due to the current in the wire shown in Figure 29.2. Rank the points A, B, and C in terms of magnitude of the magnetic field that is due to the current in just the length element $d\stackrel{\to }{s}$ shown from greatest to least.

Figure 29.2 (Quick Quiz 29.1) Where is the magnetic field due to the current element the greatest?

To determine

The rank of the points $A$, $B$ and $C$ from greatest to least in terms of magnitude of the magnetic field that is due to the current in just the length element $d\overrightarrow{\text{s}}$.

Answer to Problem 30.1QQ

The rank of the points $A$, $B$ and $C$ from greatest to least in terms of magnitude of the magnetic field that is due to the current in just the length element $d\overrightarrow{\text{s}}$ is $B>C>A$.

Explanation of Solution

Formula to calculate the magnetic field due to the current flow through a small length element is,

$\overrightarrow{\text{B}}=\frac{{\mu }_{0}I}{4\pi }{\displaystyle \int \frac{d\overrightarrow{\text{s}}\times \widehat{\text{r}}}{r^2}}$ (1)

Here,

${\mu }_0$ is the absolute permeability.

$I$ is the current flow in the conductor.

$d\overrightarrow{\text{s}}$ is the small length elements.

$r$ is the distance between the points to the conductor.

From the equation (1), it is clear that the magnetic field due to the current flow is inversely proportional to the square of the distance between the point to the small length element and directly proportional to the cross product of the vectors $d\overrightarrow{\text{s}}$ and $\widehat{\text{r}}$.

The point A lies in the direction of the small length element of the wire. So, the angle between the vectors $d\overrightarrow{\text{s}}$ and $\widehat{\text{r}}$ for point $A$ is zero. Hence, the magnitude of the magnetic field at point $A$ is zero because $\sin 0°=0$.

The distance between the point $C$ and the small length element of the wire is more than the distance between the point $B$ and the small length element of the wire. Since the field decrease as the points moves away from the small length element hence the magnitude of field at point $C$ is less than the magnitude of field at point $B$.

Thus, from the greatest to least order the magnetic field is maximum at point $B$, then at point $C$ and then at point $A$.

Conclusion:

Therefore, the rank of the points $A$, $B$ and $C$ from greatest to least in terms of magnitude of the magnetic field that is due to the current in just the length element $d\overrightarrow{\text{s}}$ are $B>C>A$.