KAHIBARO
Table of Contents
Magnetic circulation and enclosed current
Ampère's law connects electric current to the magnetic field wrapped around it. It is especially useful when the current distribution has a high degree of symmetry, such as a long straight wire, a long solenoid, or a toroid.
The law states that the line integral of the magnetic field around a closed path equals the permeability of free space times the net current enclosed by that path,
$$
\oint \mathbf{B} \cdot d\mathbf{\ell} = \mu_0 I_{\text{enc}}
$$
Here, $\mathbf{B}$ is the magnetic field, $d\mathbf{\ell}$ is a small directed element along a closed loop called an Amperian loop, $\mu_0$ is the permeability of free space, and $I_{\text{enc}}$ is the algebraic sum of currents passing through the surface bounded by the loop.
Ampère's law in magnetostatics is
$$
\oint \mathbf{B} \cdot d\mathbf{\ell} = \mu_0 I_{\text{enc}}
$$
It relates the circulation of $\mathbf{B}$ around a closed path to the net enclosed current.
What the line integral means
The expression $\oint \mathbf{B} \cdot d\mathbf{\ell}$ adds up the component of the magnetic field tangent to the chosen closed path. If the field is parallel to the path, the dot product is large. If the field is perpendicular to the path, the contribution is zero.
This means that the choice of loop matters. A clever choice makes the integral simple. In symmetric cases, the field may have constant magnitude along part or all of the loop, so the integral becomes easy to evaluate.
Orientation and sign convention
Ampère's law uses an orientation. When you choose a direction to go around the loop, the positive direction for enclosed current is set by the right hand rule. Curl the fingers of your right hand in the direction of integration around the loop. Your thumb then points in the positive direction for current through the surface.
If some current passes through the surface in the positive direction and some in the negative direction, the enclosed current is the algebraic sum.
Use the right hand rule consistently.
If the loop direction is chosen, then positive enclosed current is determined by the right hand rule for that direction.
Why symmetry is important
Ampère's law is always true for steady currents, but it is not always easy to use for finding $\mathbf{B}$. It becomes powerful when symmetry lets us know the direction of the field and whether its magnitude is constant along the loop.
Typical useful cases are shown below.
| Current distribution | Good Amperian loop | Result |
|---|---|---|
| Long straight wire | Circle centered on wire | $B$ constant on loop |
| Long solenoid | Rectangle partly inside, partly outside | Inside field approximately uniform |
| Toroid | Circle concentric with toroid | $B$ depends only on radius |
Example, long straight wire
Consider a very long straight wire carrying current $I$. By symmetry, the magnetic field forms circles around the wire, and its magnitude depends only on distance $r$ from the wire.
Choose a circular Amperian loop of radius $r$ centered on the wire. Along this loop, $\mathbf{B}$ is tangent to the circle and has constant magnitude $B$, so
$$
\oint \mathbf{B} \cdot d\mathbf{\ell} = B \oint d\ell = B(2\pi r)
$$
The enclosed current is $I$, so Ampère's law gives
$$
B(2\pi r) = \mu_0 I
$$
Therefore,
$$
B = \frac{\mu_0 I}{2\pi r}
$$
This is one of the most important results obtained from Ampère's law.
For a long straight wire,
$$
B = \frac{\mu_0 I}{2\pi r}
$$
The field decreases as $1/r$.
Example, inside a long solenoid
A long solenoid has many closely spaced turns carrying current. Its magnetic field is approximately uniform inside and very small outside. Let $n$ be the number of turns per unit length, and let each turn carry current $I$.
Choose a rectangular Amperian loop with one long side of length $L$ inside the solenoid and the other long side outside. The short sides are perpendicular to the field. Then the contributions from the short sides are zero, the outside contribution is approximately zero, and the inside contribution is $BL$.
If the loop encloses $nL$ turns, the enclosed current is
$$
I_{\text{enc}} = nLI
$$
Ampère's law gives
$$
BL = \mu_0 nLI
$$
So the magnetic field inside a long solenoid is
$$
B = \mu_0 n I
$$
Inside an ideal long solenoid,
$$
B = \mu_0 n I
$$
where $n$ is the number of turns per unit length.
Example, toroid
A toroid is like a solenoid bent into a circular ring. The magnetic field stays mostly inside the core and circles around the central axis.
Choose a circular Amperian loop of radius $r$ inside the toroid. By symmetry, $\mathbf{B}$ is tangent to the loop and has nearly constant magnitude along it. If the toroid has $N$ turns carrying current $I$, then
$$
B(2\pi r) = \mu_0 N I
$$
so
$$
B = \frac{\mu_0 N I}{2\pi r}
$$
This shows that the field inside a toroid depends on the distance from the center.
When Ampère's law is not easy to use
For irregular current distributions, Ampère's law still holds, but it may not directly give the magnetic field because the integral is hard to simplify. In those cases, other methods are often better. Ampère's law is most practical when symmetry tells us a lot about the field before we start calculating.
Ampère's law in materials
In vacuum, the constant is $\mu_0$. In magnetic materials, it is often convenient to write the field using magnetic permeability $\mu$, so simple symmetric results may appear as
$$
B = \mu n I
$$
or similar forms, depending on the situation. For beginners, the vacuum form with $\mu_0$ is the central idea.
Differential form
Ampère's law can also be written in local form using the curl of the magnetic field,
$$
\nabla \times \mathbf{B} = \mu_0 \mathbf{J}
$$
where $\mathbf{J}$ is the current density. This says that electric current creates a circulating magnetic field in the space around it.
For this chapter, the integral form is the most useful because it is the form used to solve symmetric problems.
Integral form and differential form of magnetostatic Ampère's law are
$$
\oint \mathbf{B} \cdot d\mathbf{\ell} = \mu_0 I_{\text{enc}},
\qquad
\nabla \times \mathbf{B} = \mu_0 \mathbf{J}
$$
Comparison with Gauss's law
Ampère's law for magnetism is mathematically similar to Gauss's law in electrostatics, but instead of relating flux through a closed surface to enclosed charge, it relates circulation around a closed loop to enclosed current. The physical picture is also different. Magnetic field lines tend to form closed loops around currents.
Limits of the simple form
The form
$$
\oint \mathbf{B} \cdot d\mathbf{\ell} = \mu_0 I_{\text{enc}}
$$
is valid for steady currents, meaning currents that do not change with time. In more advanced electromagnetism, Maxwell added an extra term for changing electric fields. That extension belongs to a later topic.
Summary of key results
Ampère's law gives a direct relation between circulating magnetic field and enclosed current. Its main power comes from symmetry. For a long straight wire, it gives $B = \mu_0 I/(2\pi r)$. For a long solenoid, it gives $B = \mu_0 n I$. For a toroid, it gives $B = \mu_0 N I/(2\pi r)$. The right hand rule determines the sign of enclosed current and the direction associated with the loop.