91Ó°ÊÓ

The law of Biot and Savart in Eq. ( 28.7 ) generalizes to the case of surface currents as $$ \overrightarrow{\boldsymbol{B}}=\frac{\mu_{0}}{4 \pi} \int \frac{\sigma \overrightarrow{\boldsymbol{v}} \times \hat{\boldsymbol{r}}}{r^{2}} d a $$ where \(\sigma\) is the local charge density, \(\overrightarrow{\boldsymbol{v}}\) is the local velocity, and \(d a\) is a differential area element. Re-visit Challenge Problem 28.76 and use the above equation as an alternative means to derive the magnetic field at the center of the cylinder. Use the following steps: (a) Write the charge density \(\sigma\). (b) The origin is at the center of the cylinder. What is the vector \(\vec{v}\) that points from the element with coordinates \((x, y, z)=(x, R \cos \phi, R \sin \phi)\) to the origin? (c) What is the velocity \(\overrightarrow{\boldsymbol{v}}\) of the element? (d) What is the vector product \(\overrightarrow{\boldsymbol{v}} \times \hat{\boldsymbol{r}} ?\) (e) An area element on the cylinder may be written as \(d a=R d x d \phi .\) Use this and the previously established information to write the generalized law of Biot and Savart as a double integral. Evaluate the integral to determine the magnetic field \(\vec{B}\) at the center of the cylinder. (f) Is your result consistent with your result in Challenge Problem \(28.76 ?\)

Step by step solution

01

Determining Variables

Write the charge density, \(\sigma\), as it is given by current density divided by velocity, \(J/v\). The vector pointing from the element with coordinates \((x,y,z) = (x, Rcosφ, Rsinφ)\) to the origin is \(\vec{r} = -x\hat{i} - Rcosφ\hat{j} - Rsinφ\hat{k}\) as the origin is at the center of the cylinder. The velocity (\(\vec{v}\)) of the element is given in the axial direction, thus it is \(v\hat{k}\).

02

Calculate The Vector Product \(\vec{v} \times \hat{r}\)

The vector product \(\vec{v} \times \hat{r}\) can be calculated using the right hand rule or formula for cross product which is given by \(\vec{V} \times \hat{r} = |\vec{V}||\hat{r}| sin(θ) \hat{N}\), where \(θ\) is the angle between \(\vec{V}\) and \(\hat{r}\), \(\hat{N}\) is the normal unit vector. For this case the cross product will be \(\vec{v} \times \hat{r} = vRcosφ\hat{i} + vx\hat{j}\).

03

Express the Biot-Savart Law as a double integral

We know the Biot-Savart Law in terms of surface currents as \(\vec{B} = \frac{\mu_0}{4π} \int \frac{\sigma \vec{v} \times \hat{r}}{r^2} da\). An area element on the cylinder may be written as \(da = Rdx dφ\). Substitute these values and set up the integral. The magnetic field equation becomes \(\vec{B} = \frac{\mu_0}{4π} \int_0^{2π} \int_{-L/2}^{L/2} \frac{\sigma \vec{v} \times \hat{r}}{r^2} R dx dφ\).

04

Evaluating The Integral

To evaluate the integral, break it into components. The \(i\) and \(j\) components will cancel out because of symmetry, leaving only the \(k\) component. This integral becomes \(\mu_0 vσ \int_0^{2π} \int_{-L/2}^{L/2} \frac{dx dφ}{＋\sqrt{x^2+R^2}}\). The solution to this integral will give the magnetic field \(\vec{B}\).

05

Check consistency

Compare this result with the result from Challenge Problem 28.76 to confirm consistency.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Field

Understanding the concept of a magnetic field is essential when studying electromagnetism. A magnetic field is an invisible field that exerts a force on particles that are charged and on materials that are magnetic. This field, denoted by the symbol \(\boldsymbol{B}\), is a vector field, which means it has both a magnitude and a direction at every point in space. Magnetic fields are generated by moving electric charges, such as electrons flowing through a wire, and by intrinsic magnetic moments of elementary particles associated with a fundamental quantum property called spin.

When examining the effects of magnetic fields, it's necessary to consider the right-hand rule. This rule allows us to determine the direction of the magnetic force on a moving charge. If you point your thumb in the direction of the charge's velocity, and your fingers in the direction of the magnetic field, your palm faces towards the direction of the force exerted on a positively charged particle. For negative charges, the force is in the opposite direction.

Surface Currents

Surface currents refer to the flow of electric charge across a surface area, rather than through a volume or along a line. They can be imagined as a thin sheet of charge moving over a surface, and are characterized by a surface current density \( \boldsymbol{K} \), which is a vector quantity that represents the current per unit width crossing a line on the surface. However, the magnetic field problem above is expressed in terms of surface charge density \( \boldsymbol{\( \sigma \) \), coupled with the velocity \( \boldsymbol{v} \), which gives rise to a surface current.

In a magnetic context, surface currents generate magnetic fields in the space around them, according to Ampère's law, which in some arrangements can be considered similar to Biot-Savart Law, fulfilling Maxwell's equations for magnetostatics. Understanding these currents is crucial when analyzing the magnetic fields produced by complex geometries, such as the surface of a cylinder, as in our example.

Cross Product

The cross product is a mathematical operation that can be applied to two vectors in three-dimensional space to find a third vector that is perpendicular to both of the original vectors. The magnitude of this third vector is equal to the area of the parallelogram that the original vectors span. In physics, especially in the study of electromagnetism, the cross product is used to determine the direction and magnitude of various quantities, such as the force exerted on a charged particle moving in a magnetic field.

In the Biot-Savart law equation provided, the cross product \( \boldsymbol{v} \times \boldsymbol{\hat{r}} \) defines the direction and magnitude of the infinitesimal contribution to the magnetic field from a current element. It's crucial to apply the right-hand rule to determine the direction of this cross product properly. When calculating the cross product, you will often find components that cancel out upon integrating over symmetrical geometries, simplifying the solution to the magnetic field.

Double Integral

A double integral is an integral over a two-dimensional area. When solving problems in physics, especially when calculating fields like the magnetic field due to surface currents, we use double integrals to sum up contributions from every infinitesimal area element of the surface. These contributions help us find the total effect of a distributed source.

In the context of the given problem, after setting up the Biot-Savart law with the cross product and area element included, the magnetic field at the center of the cylinder is obtained by performing a double integral over the entire surface of the cylinder. As the solution suggests, the symmetry of the system allows for significant simplifications, often leading to many terms dropping out. By executing the double integral properly, considering the specified limits of integration and the geometry of the surface, we get a precise value for the magnetic field \( \boldsymbol{B} \) at the center of the cylinder.