Problem Solving Strategy $28\mathrm{.}2$: Ampere's Law

IDENTIFY the relevant concepts:

Like Gauss$$s law for electricity, Ampere$$s law is always true but is most useful in situations where the magnetic field is symmetrical.

SET UP the problem using the following steps:

Select the integration path you will use with Ampere$$s law. If you want to determine the magnetic field at a certain point, then the path must pass through that point. The integration path has to have enough symmetry to make evaluation of the integral possible.

Determine the target variable$($s$).$ Usually this will be the magnitude of the magnetic field B$$B$$ as a function of position.

EXECUTE the solution as follows:

Carry out the integral $$B$$dl$$B$$dl$$ along your chosen integration path. If B$$B$$ is tangent to all or some portion of the integration path and has the same magnitude BB at every point, then its line integral equals BB multiplied by the length of that portion of the path. If B$$B$$ is perpendicular to some portion of the path, that portion makes no contribution to the integral.

In the integral $$B$$dl$$B$$dl$,$ B$$B$$ is always the total magnetic field at each point on the path. If no net current is enclosed by the path, the field at points on the path need not be zero, but the integral $$B$$dl$$B$$dl$$ is always zero.

Determine the current IenclIencl enclosed by the integration path. The sign of this current is given by a right$-$hand rule.

Use Ampere$$s law $$B$$dl$=$$0$Iencl$$B$$dl$=$$0$Iencl to solve for the target variable.

EVALUATE your answer:

If your result is an expression for the field magnitude as a function of position, you can check it by examining how the expression behaves in different limits$.$ What is the value of the magnetic field at a distance rr $=5\mathrm{.}73$ cmcm from the axis of the conducting tube? Recall that $0=4$$107$T$$m$/$A$0=4$$107$T$$m$/$A$.$

Express your answer numerically in teslas.