KAHIBARO
Table of Contents
Magnetic field from electric currents
The Biot-Savart law tells us how a small piece of electric current creates a magnetic field at some point in space. It is one of the basic laws used to calculate magnetic fields produced by steady currents. Here, steady means the current does not change with time.
If we know the shape of a wire, the direction of current in it, and the point where we want the magnetic field, the Biot-Savart law gives a mathematical way to add the contributions from all tiny pieces of the current path.
The idea of a current element
Imagine cutting a current-carrying wire into tiny pieces. A very small piece of wire is represented by a vector $d\vec{\ell}$. Its direction is the same as the direction of the current. Each small piece produces a small magnetic field contribution $d\vec{B}$ at the observation point.
The size of this contribution depends on three main things. It depends on the current $I$, it depends on how far the point is from the current element, and it depends on the angle between the wire element and the line joining that element to the point.
Statement of the Biot-Savart law
For a small current element, the magnetic field contribution is
$$
d\vec{B} = \frac{\mu_0}{4\pi}\frac{I \, d\vec{\ell} \times \hat{r}}{r^2}
$$
Here, $r$ is the distance from the current element to the observation point, and $\hat{r}$ is a unit vector pointing from the current element toward that point. The symbol $\times$ is the cross product, so the magnetic field direction is perpendicular to both $d\vec{\ell}$ and $\hat{r}$.
The magnitude form is
$$
dB = \frac{\mu_0}{4\pi}\frac{I \, d\ell \sin\theta}{r^2}
$$
where $\theta$ is the angle between $d\vec{\ell}$ and the vector toward the observation point.
To find the total magnetic field, we integrate over the whole current distribution:
$$
\vec{B} = \frac{\mu_0}{4\pi}\int \frac{I \, d\vec{\ell} \times \hat{r}}{r^2}
$$
Biot-Savart law for a steady current:
$$
d\vec{B} = \frac{\mu_0}{4\pi}\frac{I \, d\vec{\ell} \times \hat{r}}{r^2}
$$
Total field:
$$
\vec{B} = \frac{\mu_0}{4\pi}\int \frac{I \, d\vec{\ell} \times \hat{r}}{r^2}
$$
Important facts:
$$
dB \propto I, \qquad dB \propto \frac{1}{r^2}, \qquad dB \propto \sin\theta
$$
Meaning of the symbols
The constant $\mu_0$ is the permeability of free space. In SI units,
$$
\mu_0 = 4\pi \times 10^{-7}\ \text{T m/A}
$$
The magnetic field $\vec{B}$ is measured in tesla, $\text{T}$. Current $I$ is measured in amperes, $\text{A}$.
The law is very similar in spirit to Coulomb's law in electrostatics, but there is one major difference. The magnetic field depends on direction through the cross product. Because of this, magnetic fields curl around currents instead of pointing directly away from them.
Direction of the magnetic field
The direction of $d\vec{B}$ is given by the right hand rule for the cross product. Point the fingers of your right hand along $d\vec{\ell}$. Curl them toward $\hat{r}$. Your thumb points in the direction of $d\vec{B}$.
This directional rule is very important. Even when different current elements have the same distance from the point, their field directions may be different. In many problems, some parts cancel while others add.
Dependence on angle
The factor $\sin\theta$ shows that the magnetic field is strongest when the current element is perpendicular to the line joining it to the observation point, so $\theta = 90^\circ$. Then $\sin\theta = 1$.
If the observation point lies directly along the line of the current element, then $\theta = 0^\circ$ or $180^\circ$, so $\sin\theta = 0$. In that case, that current element contributes no magnetic field at that point.
Using symmetry
The Biot-Savart law can be used for almost any wire shape, but the calculation may be difficult. It becomes especially useful when the geometry is simple and symmetric. Common examples include a long straight wire, a circular loop, or the center of a current arc.
In these cases, symmetry helps us understand which field components cancel and which survive.
Field of a long straight wire
For an infinitely long straight wire carrying current $I$, the magnetic field at distance $r$ from the wire is
$$
B = \frac{\mu_0 I}{2\pi r}
$$
The field forms circles around the wire. Its direction is given by the right hand grip rule. Point your thumb in the direction of current, and your fingers curl in the direction of the magnetic field.
For a long straight wire:
$$
B = \frac{\mu_0 I}{2\pi r}
$$
The field decreases as $\frac{1}{r}$, not as $\frac{1}{r^2}$, because the contributions from the whole wire are added by integration.
Field at the center of a circular loop
For a circular loop of radius $R$ carrying current $I$, the magnetic field at the center is
$$
B = \frac{\mu_0 I}{2R}
$$
Every current element on the loop is the same distance from the center, and all contributions point in the same axial direction. That is why the field adds neatly.
If the loop has $N$ turns, the field becomes
$$
B = \frac{\mu_0 N I}{2R}
$$
Field due to a circular arc
A useful result comes from part of a circle. If a wire forms an arc of angle $\phi$, measured in radians, and radius $R$, then the magnetic field at the center of the arc is
$$
B = \frac{\mu_0 I \phi}{4\pi R}
$$
This includes several special cases. If $\phi = 2\pi$, the arc is a full circle, and we recover
$$
B = \frac{\mu_0 I}{2R}
$$
If $\phi = \pi$, the wire is a semicircle, and
$$
B = \frac{\mu_0 I}{4R}
$$
Why integration is needed
A single current element gives only a tiny contribution $d\vec{B}$. Real wires contain many such elements, each at a different distance and direction relative to the point of interest. So we add them continuously using an integral.
In practice, solving a Biot-Savart problem usually follows this pattern. First choose a small current element $d\vec{\ell}$. Then write the vector from that element to the observation point. Then determine the cross product direction and magnitude. Finally integrate over the entire wire.
Comparison of common results
| Current shape | Magnetic field at chosen point |
|---|---|
| Infinitely long straight wire, distance $r$ | $B = \dfrac{\mu_0 I}{2\pi r}$ |
| Circular loop, center, radius $R$ | $B = \dfrac{\mu_0 I}{2R}$ |
| $N$ circular turns, center | $B = \dfrac{\mu_0 N I}{2R}$ |
| Circular arc, center, angle $\phi$ | $B = \dfrac{\mu_0 I \phi}{4\pi R}$ |
Limitations of the law
The Biot-Savart law in this form is used for steady currents. If currents change with time, electromagnetic effects become more complicated, and this simple form is not enough by itself.
Also, while the law is very general, some problems are hard to solve directly. In highly symmetric situations, another method can be easier, but the Biot-Savart law remains a fundamental starting point for understanding how currents create magnetic fields.
Key idea:
The Biot-Savart law gives the magnetic field produced by current elements, and the total field is found by integrating over the whole current path.
Direction is not optional. Because the law uses a cross product, always determine the field direction with the right hand rule.