Question

Current density in a cylindrical wire is varying with radial distance r as $J = \frac{J_0}{R^2}r(R-r)$, where R is radius of wire. If B is magnetic field due to this wire, then choose the incorrect options.

• A. Maximum magnetic field will be at a distance $\frac{8R}{9}$
• B. Maximum magnetic field will be at a distance R.
• C. Maximum magnetic field will be $\frac{\mu_0J_0R}{12}$
• D. Maximum magnetic field will be $\frac{64\mu_0J_0R}{729}$

Answer 1

Answer: (B), (C)

B, C

Answer 2

The current density in the cylindrical wire is given by $J = \frac{J_0}{R^2}r(R-r)$, where $R$ is the radius of the wire and $r$ is the radial distance from the axis.

We use Ampere's Law to find the magnetic field $B(r)$. For a circular Amperian loop of radius $r$ centered on the axis, Ampere's Law is $B(r) \cdot 2\pi r = \mu_0 I_{enc}(r)$, where $I_{enc}(r)$ is the current enclosed by the loop.

Case 1: Magnetic field inside the wire ($r \le R$)

The enclosed current is $I_{enc}(r) = \int_0^r J(r') dA = \int_0^r \frac{J_0}{R^2}r'(R-r') (2\pi r' dr')$.

$I_{enc}(r) = \frac{2\pi J_0}{R^2} \int_0^r (Rr'^2 - r'^3) dr' = \frac{2\pi J_0}{R^2} \left[ \frac{Rr'^3}{3} - \frac{r'^4}{4} \right]_0^r = \frac{2\pi J_0}{R^2} \left( \frac{Rr^3}{3} - \frac{r^4}{4} \right)$.

Using Ampere's Law, $B(r) \cdot 2\pi r = \mu_0 \frac{2\pi J_0}{R^2} \left( \frac{Rr^3}{3} - \frac{r^4}{4} \right)$.

$B(r) = \frac{\mu_0 J_0}{R^2} \left( \frac{Rr^2}{3} - \frac{r^3}{4} \right)$ for $0 \le r \le R$.

To find the maximum magnetic field inside the wire, we differentiate $B(r)$ with respect to $r$ and set it to zero.

$\frac{dB}{dr} = \frac{\mu_0 J_0}{R^2} \left( \frac{2Rr}{3} - \frac{3r^2}{4} \right)$.

Setting $\frac{dB}{dr} = 0$ gives $\frac{2Rr}{3} - \frac{3r^2}{4} = 0$, which factors as $r \left( \frac{2R}{3} - \frac{3r}{4} \right) = 0$.

This gives $r=0$ or $\frac{2R}{3} = \frac{3r}{4}$, so $r = \frac{8R}{9}$.

The value $r = \frac{8R}{9}$ is within the wire ($0 < \frac{8R}{9} < R$).

We can check the second derivative or the values at critical points and boundaries.

$B(0) = 0$.

$B(R) = \frac{\mu_0 J_0}{R^2} \left( \frac{R(R)^2}{3} - \frac{R^3}{4} \right) = \frac{\mu_0 J_0}{R^2} \left( \frac{R^3}{3} - \frac{R^3}{4} \right) = \frac{\mu_0 J_0}{R^2} \frac{R^3}{12} = \frac{\mu_0 J_0 R}{12}$.

$B\left(\frac{8R}{9}\right) = \frac{\mu_0 J_0}{R^2} \left( \frac{R}{3}\left(\frac{8R}{9}\right)^2 - \frac{1}{4}\left(\frac{8R}{9}\right)^3 \right) = \frac{\mu_0 J_0}{R^2} \left( \frac{R}{3}\frac{64R^2}{81} - \frac{1}{4}\frac{512R^3}{729} \right)$

$B\left(\frac{8R}{9}\right) = \frac{\mu_0 J_0}{R^2} \left( \frac{64R^3}{243} - \frac{128R^3}{729} \right) = \mu_0 J_0 R \left( \frac{64}{243} - \frac{128}{729} \right) = \mu_0 J_0 R \left( \frac{192 - 128}{729} \right) = \frac{64\mu_0 J_0 R}{729}$.

Comparing $B\left(\frac{8R}{9}\right)$ and $B(R)$: $\frac{64}{729} \approx 0.08779$ and $\frac{1}{12} \approx 0.08333$. Since $\frac{64}{729} > \frac{1}{12}$, the maximum magnetic field inside the wire occurs at $r = \frac{8R}{9}$.

Case 2: Magnetic field outside the wire ($r > R$)

The total current flowing through the wire is $I_{total} = I_{enc}(R) = \frac{2\pi J_0}{R^2} \frac{R^4}{12} = \frac{\pi J_0 R^2}{6}$.

For $r > R$, $B(r) \cdot 2\pi r = \mu_0 I_{total} = \mu_0 \frac{\pi J_0 R^2}{6}$.

$B(r) = \frac{\mu_0 J_0 R^2}{12r}$ for $r > R$.

This function decreases as $r$ increases for $r > R$. The maximum value for $r \ge R$ occurs at $r=R$, which is $B(R) = \frac{\mu_0 J_0 R}{12}$.

Comparing the maximum value inside the wire $B\left(\frac{8R}{9}\right) = \frac{64\mu_0 J_0 R}{729}$ with the maximum value outside (which occurs at the surface) $B(R) = \frac{\mu_0 J_0 R}{12}$, we found that $B\left(\frac{8R}{9}\right) > B(R)$.

Thus, the overall maximum magnetic field occurs at $r = \frac{8R}{9}$ and its value is $\frac{64\mu_0 J_0 R}{729}$.

Now let's evaluate the options:

A Maximum magnetic field will be at a distance $\frac{8R}{9}$. This is correct.

B Maximum magnetic field will be at a distance R. This is incorrect. The maximum is at $r = \frac{8R}{9}$.

C Maximum magnetic field will be $\frac{\mu_0J_0R}{12}$. This is incorrect. This is the value at $r=R$, not the maximum value.

D Maximum magnetic field will be $\frac{64\mu_0J_0R}{729}$. This is correct. This is the value at $r = \frac{8R}{9}$.

The question asks for the incorrect options. Options B and C are incorrect.