Question

Figure 29-29 gives, as a function of radial distance r, the magnitude Bof the magnetic field inside and outside four wires (a, b, c, and d), each of which carries a current that is uniformly distributed across the wire’s cross section. Overlapping portions of the plots (drawn slightly separated) are indicated by double labels. Rank the wires according to (a) radius, (b) the magnitude of the magnetic field on the surface, and (c) the value of the current, greatest first. (d) Is the magnitude of the current density in wire agreater than, less than, or equal to that in wire c?

Short answer

(a) The ranking of wires according to radius is $c>a>d>b$.

(b) The ranking of wires according to the magnitude of magnetic field on the surface is$a>c>b>d$.

(c) The ranking of wires according to the value of current is$a=c>b=d$.

(d) The magnitude of the current density in wire a is greater than that in wire c.

Step-by-step solution

Given

• Figure 29-29.
• Current is uniformly distributed across the wire’s cross-section.

Determining the concept

Rank the wires according to the radius by analyzing the given figure. Rank the wires according tothe magnitude of the magnetic field on the surface using the formula for the magnetic field in the wire. Also, rank them according to the value of current using the formula for the magnetic field outside the wire. Using the formula for current density, determine whether the magnitude of the current density in wirea is greater than, less than, or equal to that in wire c.

Formulae are as follow:

$$\begin{gathered} B=\left(\frac{{\mu }_{0}i}{2\pi R^2}\right)r \\ B=\frac{{\mu }_{0}i}{2\pi r} \\ J=i/A \end{gathered}$$

Where, B is the magnetic field, R is the radius, iis current, 𝛍is the permeability, J is the current density, A is area.

(a) Determining the rank of wires according to the radius

Magnetic field outside the wires is inversely proportional to the distance. So, the distance from which the field starts decreasing is equal to the radius.

Analyzing the given figure, it can be concluded that the ranking of wires according to the radius is, $c>a>d>b$.

Hence, the ranking of wires according to radius is.$c>a>d>b$

(b) Determining the rank of wires according to the magnitude of magnetic field on the surface

In the graph it can be seen, that magnetic field increases linearly with r until it reaches maximum. This represents the magnetic field inside the wire. The magnetic field inside the wire is given by,

$B=\left(\frac{{\mu }_{0}i}{2\pi R^2}\right)r$

The maximum value of B is at the surface.

Hence, analyzing the graph gives

Hence, the ranking of wires according to the magnitude of magnetic field on the surface is,$a>c>b>d$ .

(c) Determining the rank of wires according to the value of current

Magnetic field outside the wire is given by,

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

Hence,

$$\begin{gathered} i=\frac{2\pi rB}{{\mu }_0} \\ i\propto rB \end{gathered}$$

Thus, from the given figure conclude that the ranking of wires according to the value of current is,$a=c>b=d$ .

Hence, the ranking of wires according to the value of current is$a=c>b=d$

(d) Determining whether the magnitude of the current density in the wire a is greater than, less than, or equal to that in wire c

Current is uniformly distributed across the wire’s cross section. Hence, the current density is given by,

$J=i/A$

In part a) the radius of c is greater than that of a. Hence, area of c will be greater than that of wire a.

According to the part c) current is same for both wires a andc.

Hence, the magnitude of the current density in wire a is greater than that in wire c.

Therefore, analyzing the graph between magnetic field inside and outside the wire, and radial distance, rank the wires according to the radius, magnitude of magnetic field on the surface, value of current, and current density.