Question

The current density in a cylindrical conductor of radius \(R\) varies as \(J(r)=J_{0} e^{-r / R}(\) in the region from zero to \(R\) ). Express the magnitude of the magnetic field in the regions \(rR .\) Produce a sketch of the radial dependence, \(B(r)\).

Short answer

Answer: The radial dependence of the magnetic field B(r) in a cylindrical conductor with a given current density J(r) shows that the magnetic field decreases as we move away from the center of the conductor (r=0) and goes to zero when r goes to infinity. In the region r < R, the magnetic field decreases rapidly for small values of r and more slowly as r increases. In the region r > R, the magnetic field decreases as 1/r³. The overall shape of the radial dependence of the magnetic field B(r) resembles a sigmoid function with a higher decrease rate in the region r < R and a lower decrease rate in the region r > R.

Step-by-step solution

Recall Ampere's Law and Biot-Savart Law

Ampere's Law states that the line integral of magnetic field B around any closed path is equal to the product of the permeability of free space (μ₀) and the enclosed current (I): \(\oint \textbf{B} \cdot d\textbf{l} = \mu_{0} I\). Biot-Savart Law is used to find the magnetic field created by a steady current distribution in space.

Calculating the enclosed current

To find the magnetic field B(r), we need to compute the enclosed current in the regions r < R and r > R. The enclosed current can be found by integrating the given current density \(J(r)=J_{0} e^{-r / R}\) over the area A. For the region r < R: \(I(r) = \int_{0}^{2\pi} \int_{0}^{r} J(r') r' dr' d\theta = \int_{0}^{2\pi} \int_{0}^{r} J_{0} e^{-r' / R} r' dr' d\theta\) For the region r > R: \(I(R) = \int_{0}^{2\pi} \int_{0}^{R} J(r') r' dr' d\theta = \int_{0}^{2\pi} \int_{0}^{R} J_{0} e^{-r' / R} r' dr' d\theta\)

Applying Ampere's Law

Now we apply Ampere's Law to find the magnetic field B(r) in both regions. For the region r < R: \(B(r) \cdot 2\pi r = \mu_{0} I(r)\) \(B(r) = \frac{\mu_{0}}{2\pi r} I(r)\) For the region r > R: \(B(r) \cdot 2\pi r = \mu_{0} I(R)\) \(B(r) = \frac{\mu_{0}}{2\pi r} I(R)\)

Calculation of B(r)

Insert the expressions for I(r) and I(R) into the expressions for B(r) in both regions, and we get: For the region r < R: \(B(r) = \frac{\mu_{0} J_{0}}{2} \frac{1-e^{-r/R}}{r}\) For the region r > R: \(B(r) = \frac{\mu_{0} J_{0} R^2}{2r^3} (1-e^{-1})\)

Sketch the radial dependence B(r)

From the expressions for B(r) in both regions, we see that the magnetic field is decreasing as we move away from the center of the conductor (r = 0), and it goes to zero when r goes to infinity. In the region r < R, the magnetic field decreases rapidly for small values of r and then more slowly as r increases. In the region r > R, the magnetic field decreases as \(1/r^3\). Overall, the radial dependence of the magnetic field B(r) looks like a sigmoid function with a higher decrease rate in the region r < R and a lower decrease rate in the region r > R.

Key concepts

Ampere's Law

Understanding the behavior of magnetic fields in conductors is heavily reliant on Ampere's Law, one of the fundamental rules of electromagnetism that relates the magnetic field around a closed loop to the electrical current passing through it.

The elegance of Ampere's Law is encapsulated by its simple, yet powerful mathematical expression: \(oint \textbf{B} \bullet d\textbf{l} = \[0pt] \mu_{0} I\)\). Here, \(\textbf{B}\) represents the magnetic field, \(d\textbf{l}\) is an infinitesimally small element of the loop, \(I\) is the current enclosed by the loop, and \(\mu_{0}\) is the permeability of free space.

This law is particularly useful because it allows us to calculate the magnetic field generated by any kind of current running through a conductor, as long as we can identify an appropriate path around which to integrate. In the exercise at hand, we exploit Ampere's Law to deduce the magnetic field inside and outside a cylindrical conductor, with a unique current distribution.

Biot-Savart Law

In addition to Ampere's Law, the Biot-Savart Law plays a crucial role in the study of magnetic fields, especially when dealing with the magnetic field created by a steady current distribution. The Biot-Savart Law provides a relationship to calculate the magnetic field produced at a point due to a small segment of current-carrying conductor.

The law is expressed by the formula: \( d\textbf{B} = \frac{\mu_{0}}{4\pi} \frac{Id\textbf{l} \times \hat{\textbf{r}}}{r^2} \), where \( d\textbf{B} \) is the differential magnetic field generated by the current element \(Id\textbf{l}\), \(\hat{\textbf{r}}\) is the unit vector from the current element to the point of observation, and \(r\) is the distance to the observation point.

While Biot-Savart Law is more generic and can be applied in situations where Ampere's Law may not be directly applicable, in the context of the cylindrical conductor problem, it reaffirms the symmetry that allows us to use Ampere's Law effectively.

Current Density

The concept of current density is pivotal in predicting how electric current is distributed within conductors. Current density, denoted as \(\mathbf{J}\), is a vector quantity that describes the flow of electric charge per unit area across a given cross-sectional area. In mathematical terms, it is defined as \(\mathbf{J} = \frac{dI}{dA}\), where \(dI\) is the infinitesimal current flowing through an infinitesimal area \(dA\).

In our specific exercise, the current density is given by a decaying exponential function \(J(r)=J_{0} e^{-r / R}\) where \(J_{0}\) is the maximum current density at the center of the conductor \(r = 0\) and decreases as one moves outward. This distribution leads to a non-uniform magnetic field within the conductor that varies with the radial distance \(r\) from the center, thereby complicating the calculation of the total current enclosed by a loop used in Ampere's Law. The current density helps us calculate the total current enclosed within a certain radius which is essential for finding the magnetic field using both Ampere's and Biot-Savart's Laws.

Cylindrical Conductor

The geometry of the conductor in question greatly influences the magnetic field's configuration. A cylindrical conductor, as discussed in the example, is a common and well-understood shape where cylindrical symmetry simplifies the analysis of the magnetic field. When examining the magnetic field inside and outside a cylindrical conductor, the symmetry allows for a simplification where any path used in Ampere's Law can be a circle centered on the axis of the cylinder.

Thanks to this symmetry, the magnetic field at a distance \(r\) from the center of the cylinder, for the given current density \(J(r)\), is calculated in a straightforward manner as long as calculations adhere to the boundaries of the conductor, defined by the radius \(R\). Identifying these boundaries correctly and accounting for them in the calculations for different regions, namely \(rR\), enables us to find the exact expression for the magnetic field in each corresponding region of this cylindrical space.