Question

The current density in a cylindrical conductor of radius \(R\) varies as \(J(r)=J_{0} r / R\) (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

Question: Determine the expressions for the magnetic field inside and outside a cylindrical conductor with a current density that varies linearly with the distance from the center. Then, describe the radial dependence of the magnetic field. Answer: The magnetic field inside the cylindrical conductor is given by the expression \(B(r) = \frac{\mu_0 J_0}{2R} r\) for \(r < R\), where \(r\) is the radial distance from the center, \(\mu_0\) is the permeability of free space, \(J_0\) is the maximum current density, and \(R\) is the radius of the conductor. Outside the conductor, the magnetic field is given by the expression \(B(r) = \frac{\mu_0 J_0 R^2}{3r}\) for \(r > R\). The radial dependence of the magnetic field shows that the magnetic field increases linearly with the radial distance inside the conductor and decreases as the inverse of the radial distance outside the conductor.

Step-by-step solution

Set up the problem

We have a cylindrical conductor of radius \(R\) with a current density \(J(r) = J_{0}r/R\). Let's assume a cylindrical surface of radius \(r (< R)\) inside the conductor, and using Ampere's law in a closed loop, integrate the current density through this surface to calculate the magnetic field inside the conductor.

Use Ampere's law inside the conductor

We can write the Ampere's law for the cylinder with radius \(r\) as: $$\oint B \, d \ell = \mu_0 I_{enc}$$ The enclosed current \(I_{enc}\) can be found by integrating the current density over the area of the cylinder with radius \(r\): $$I_{enc} = \int_0^r \int_0^{2\pi} J \, r d\phi dr = \int_0^r \int_0^{2\pi} \frac{J_0 r^2}{R} d\phi dr$$ Now, integrate to find the enclosed current: $$I_{enc} = J_0\int_{0}^{r} \frac{r^2}{R} 2\pi dr = \frac{2\pi J_0}{R} \int_{0}^{r} r^2 dr = \frac{2\pi J_0}{R} \frac{r^3}{3}$$ Now, put the enclosed current back into Ampere's law equation: $$B \oint d \ell = \mu_0 \left(\frac{2\pi J_0}{R} \frac{r^3}{3}\right)$$ $$B(2\pi r) = \mu_0 \left(\frac{2\pi J_0}{R} \frac{r^3}{3}\right) \implies B(r) = \frac{\mu_0 J_0}{2R} r\quad (r<R)$$

Use Biot-Savart law outside the conductor

For the point \(r>R\), we can use the Biot-Savart law to find the magnetic field: $$d\vec{B} = \frac{\mu_0}{4\pi}\frac{I d\vec{l} \times \vec{r'}}{|\vec{r'}|^3}$$ Considering the symmetry of the problem, the magnetic field outside the conductor will be the same as for a wire carrying the total current. The total current can be calculated by integrating the current density over the entire area of the conductor: $$I_{total} = \int_{0}^{R} \int_{0}^{2\pi} J \, r d\phi dr = \int_{0}^{R} \int_{0}^{2\pi} \frac{J_{0} r^2}{R} d\phi dr = \frac{2\pi J_0}{R} \int_{0}^{R} r^2 dr = \frac{2\pi J_0}{R} \frac{R^3}{3}$$ The magnetic field outside the conductor is given by: $$B(r) = \frac{\mu_0 I_{total}}{2\pi r}\quad (r>R)$$ Substituting the total current, we get: $$B(r) = \frac{\mu_0 \left(\frac{2\pi J_0 R^2}{3}\right)}{2\pi r}\quad (r>R)$$ Hence, for points outside the conductor: $$B(r) = \frac{\mu_0 J_0 R^2}{3r}\quad (r>R)$$

Sketch the radial dependence of the magnetic field

Now we have expressions for the magnetic field both inside and outside the conductor: $$B(r) = \begin{cases} \frac{\mu_0 J_0}{2R} r &\text{ for } r < R\\ \frac{\mu_0 J_0 R^2}{3r} &\text{ for } r > R \end{cases}$$ To sketch the radial dependence of the magnetic field (\(B(r)\)), we can see that the magnetic field increases linearly with \(r\) inside the conductor and decreases as \(1/r\) outside the conductor.

Key concepts

Understanding Ampere's Law

A fundamental tool in understanding magnetic fields is Ampere's Law, named after André-Marie Ampère. Ampere's Law relates the integrated magnetic field around a closed loop to the electric current passing through the loop. In mathematical terms, it is represented as \[\oint B \. d\ell = \mu_0 I_{\text{enc}}\]where \(B\) is the magnetic field, \(d\ell\) is a differential length element of the closed path, \(\mu_0\) is the permeability of free space, and \(I_{\text{enc}}\) is the current enclosed by the path.
For our particular problem, we apply Ampere's Law to a cylindrical surface inside the conductor to determine the magnetic field in that region. The elegance of Ampere's Law lies in its ability to simplify the calculations for systems with high symmetry, such as our cylindrical conductor where we easily inferred the linear relationship between the magnetic field inside the conductor and the radius, \( r \).

Why High Symmetry Matters

High symmetry in problems like these means that the magnetic field is uniform along the integration path chosen—a circle of radius \( r \) in our case—allowing us to take the magnetic field outside the integral. This greatly simplifies the problem, as the path integral of a differential length around a circle is just the circumference of the circle, \(2\pi r\). As a result, Ampere's Law provides a straightforward formula for the magnetic field within symmetric systems that contain steady currents.

Current Density and its Implications

Current density is a measure of the electric current per unit area of cross-section, denoted by \(J\). In our cylindrical conductor problem, the current density is not constant and varies with the radius according to the relationship \[J(r) = J_0 \frac{r}{R}\]where \(J_0\) is a constant, and \(r\) is the distance from the center of the conductor. This radial dependence means that the further you are from the center, the higher the current density, up to the limit \(R\), the radius of the conductor.
Understanding current density is pivotal for solving magnetic field problems using Ampere's Law. We calculate the enclosed current by integrating the current density over the area through which it flows. In terms of physical implications, a varying current density like the one presented affects the distribution of the magnetic field within the material. This results in the linearly increasing magnetic field inside the conductor, as demonstrated in the solution.

Different Current Distributions

It's important to note that various current distributions give rise to different magnetic field patterns. For instance, a constant current density would result in a different magnetic field behavior compared to our current varying one. This highlights the importance of accurate current density expressions for precise magnetic field calculations in various electromagnetic applications.

Applying the Biot-Savart Law

The Biot-Savart Law is another crucial equation in magnetostatics that allows us to calculate the magnetic field generated by a steady current. It is especially useful for calculating the magnetic field in configurations with less symmetry than what is required for Ampere's Law. The Biot-Savart Law is given by\[d\vec{B} = \frac{\mu_0}{4\pi}\frac{I d\vec{l} \times \vec{r'}}{|\vec{r'}|^3}\]where \(d\vec{B}\) is the infinitesimal magnetic field due to an infinitesimal segment of current \(I d\vec{l}\), and \(\vec{r'}\) is the position vector from the current element to the point of observation.
In the context of our cylindrical conductor, outside of it, we treat the total current as if it were concentrated along the conductor's axis. This simplification is possible because of the symmetry of the cylindrical shape. The Biot-Savart Law then provides the magnetic field at a point of distance \(r>R\) away from the conductor and shows how it inversely decreases with distance. The overall decrease of the magnetic field with distance from the conductor is a general feature of magnetic fields in space and is vital for calculating the influence of magnetic sources in diverse situations, from engineering applications to understanding natural phenomena.