Q4P - Fundamentals Of Physics

Question

Two wires, parallel to a z-axis and a distance 4r apart, carry equal currents i in opposite directions, as shown in Figure. A circular cylinder of radius r and length L has its axis on the z-axis, midway between the wires. Use Gauss’ law for magnetism to derive an expression for the net outward magnetic flux through the half of the cylindrical surface above the x axis. (Hint: Find the flux through the portion of the xz plane that lies within the cylinder.)

Step-by-Step Solution

Answer

An expression for the net outward magnetic flux through half of the cylindrical surface above the x-axis is, ${\Phi }_B\left(s\right)=\frac{{\mu }_{0}iL}{\pi }in 3$.

Step 1: Identification of the given data

Two wires, parallel to Z the axis and distance 4r apart carry equal current i in opposite directions.

Hint: The flux through the portion of the xz plane that lies within the cylinder.

Step 2: Determining the concept

Applying the Gauss law for magnetism, the flux through the S₁ portion, i.e., the upside portion, is equal to the flux through the S₂, i.e., downside portion, of the XY plane that lies within the cylinder in terms of magnetic field and length of a circular cylinder. Then using the formula for the magnetic field due to the current passing through an infinite straight wire, the expression can be found for the net outward magnetic flux through half of the cylindrical surface above the x-axis.
 

The formula is as follows:

${\mathit{\Phi }}_{\mathbf{B}}\mathbf{=}\mathbf{\oint }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}\mathbf{.}\mathit{d}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{A}}\mathbf{.}$

Where,

${\Phi }_B$ is the Magnetic flux

B is the magnetic induction

A is the area

Step 3: Determining an expression for the net outward magnetic flux through half of the cylindrical surface above the x- axis

Applying the Gauss law for magnetism, the flux through the S₁ portion, i.e., the upside portion, is equal to the flux through the S₂, i.e., downside portion, of the xz plane.

According to Gauss law,

${\Phi }_{\mathrm{B}}=\oint \stackrel{\rightharpoonup }{\mathrm{B}}.d\stackrel{\rightharpoonup }{\mathrm{A}}.$

In this case, the magnetic flux through S₁ portion is ${\Phi }_B\left(S_1\right)=\underset{-r}{\overset{r}{\int }}B\left(x\right)Ldx$.

Similarly, the magnetic flux through portion S₁ is ${\Phi }_B\left(S_2\right)=\underset{-r}{\overset{r}{\int }}B\left(x\right)Ldx$.

But the magnitude of the magnetic field is the same in both portions. So,

${\Phi }_B\left(S_1\right)={\Phi }_B\left(S_2\right)$

The net magnetic flux is,

${\Phi }_B\left(S\right)=\underset{-r}{\overset{r}{\int }}B\left(x\right)Ldx$

where the magnetic field due to the current passing through an infinite straight wire is,

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

Here, the r is 2r - x

So that the magnetic field is, $B=\frac{{\mu }_{0}i}{2\pi \left(2r-x\right)}$.

Hence,

$\begin{array}{rcl}{\Phi }_B\left(s\right)& =& 2\underset{-r}{\overset{r}{\int }}\frac{{\mu }_{0}i}{2\pi \left(2r-x\right)}Ldx\\ {\Phi }_B\left(s\right)& =& \frac{{\mu }_{0}iL}{\pi }\underset{-r}{\overset{r}{\int }}\frac{dx}{\left(2r-x\right)}\\ {\Phi }_B\left(s\right)& =& \frac{{\mu }_{0}iL}{\pi }{\left[-In\left(2r-x\right)\right]}_{-r}^r\\ {\Phi }_B\left(s\right)& =& \frac{{\mu }_{0}iL}{\pi }\left[In\left(3r\right)-In\left(r\right)\right]\\ {\Phi }_B\left(s\right)& =& \frac{{\mu }_{0}iL}{\pi }\left[In\left(\frac{3r}{r}\right)\right]\\ {\Phi }_B\left(s\right)& =& \frac{{\mu }_{0}iL}{\pi }In3\end{array}$

Hence the expression for the net outward magnetic flux through half of the cylindrical surface above the x-axis is, ${\Phi }_B\left(s\right)=\frac{{\mu }_{0}iL}{\pi }In3$.

Using the concept of Gauss law for magnetism, the expression for the net outward magnetic flux through half of the cylindrical surface above the x-axis can be found.