12.3 Magnetic Fields of Current-Carrying Wires and the Biot-Savart Law

TLDR

The Biot-Savart law lets you calculate the magnetic field a current creates at a point in space using $d\vec{B}=\frac{\mu_{0}}{4\pi}\frac{I(d\vec{\ell}\times\hat{r})}{r^{2}}$. For AP Physics C: E&M, you mainly use it for symmetric setups like the center or axis of a circular loop and the perpendicular bisector of a straight wire, and you also need to find the force a magnetic field exerts on a current-carrying wire.

The Biot-Savart law gives the magnetic field a current creates, and questions ask you to sketch the field patterns and predict by what factor the field changes when you change the current, radius, or distance. Use the functional dependence, such as the 1/r falloff for a long straight wire, to find those factors.

Fields From Current and Forces on Wires

• The Biot-Savart law gives the field from a current element: $d\vec{B}=\frac{\mu_{0}}{4\pi}\frac{I(d\vec{\ell}\times\hat{r})}{r^{2}}$, and you integrate over the conductor for the total field.
• Field lines around a straight wire are concentric circles; the field is tangent to those circles with no radial or axial component. Use the right-hand grip rule for direction.
• Memorize the two derived results you are expected to use: $B_{\text{center of loop}}=\frac{\mu_{0}I}{2R}$ and the long straight wire field $B=\frac{\mu_{0}I}{2\pi r}$.
• Quantitative Biot-Savart work is limited to symmetric cases: the perpendicular bisector of a straight conductor, the central axis of a circular loop, or the center of a circular arc segment.
• A magnetic field exerts a force on a current-carrying wire: $\vec{F}_{B}=\int I(d\vec{\ell}\times\vec{B})$, which becomes $F=ILB\sin\theta$ for a straight wire in a uniform field.
• Direction always comes from the cross product and the right-hand rule, so reversing the current reverses the field.

Biot-Savart Law

The Biot-Savart law tells you the magnetic field a small piece of current-carrying wire creates at a point in space. It connects the field to the current's size, direction, and distance from the source.

The differential form is:

$d \vec{B}=\frac{\mu_{0}}{4 \pi} \frac{I(d\vec{\ell} \times \hat{r})}{r^{2}}$

Each term has a clear meaning:

• $\mu_0 = 4\pi \times 10^{-7}$ T·m/A is the permeability of free space
• $I$ is the current through the wire (in amperes)
• $d\vec{\ell}$ is a small length vector of the wire segment, pointing in the direction of current flow
• $\hat{r}$ is the unit vector pointing from the wire segment to the field point
• $r$ is the distance from the wire segment to the field point

The cross product $d\vec{\ell} \times \hat{r}$ is what makes the field perpendicular to both the current direction and the line from the source to the point. To get the total field, you integrate each contribution over the whole conductor:

$\vec{B} = \frac{\mu_{0}}{4 \pi} \int \frac{I(d\vec{\ell} \times \hat{r})}{r^{2}}$

For a circular loop, applying the Biot-Savart law at the center gives a clean result:

$B_{\text{center of loop}} = \frac{\mu_{0} I}{2 R}$

where $R$ is the loop radius.

Magnetic Field Vectors Around a Wire

The field around a straight current-carrying wire follows a clear pattern that lines up with the right-hand rule. Picturing it correctly helps with both direction questions and integration setups.

For a straight wire, the magnetic field vectors:

• Form concentric circles centered on the wire
• Are tangent to those circles at every point
• Have no component pointing toward, away from, or parallel to the wire
• Follow the right-hand grip rule: point your thumb in the direction of conventional current, and your fingers curl in the direction of the field

Reversing the current reverses every field vector. That is a direct result of the cross product in the Biot-Savart law.

Setting Up Biot-Savart Calculations

Calculating fields with the Biot-Savart law usually means integrating, which gets hard for messy shapes. The exam keeps you in symmetric cases where the math stays manageable.

The general process is:

1. Break the wire into infinitesimal segments $d\vec{\ell}$
2. Find each segment's contribution using the Biot-Savart law
3. Integrate those contributions over the whole conductor, watching for symmetry that cancels components

For a long, straight wire, the field at a perpendicular distance $r$ is:

$B = \frac{\mu_0 I}{2\pi r}$

The field weakens as $1/r$, so it falls off as you move away from the wire. You can derive this same result more directly with Ampère's law in Topic 12.4, but the Biot-Savart approach reinforces where it comes from.

Force on Current-Carrying Wires

When a current-carrying wire sits in an external magnetic field, the field pushes on it. This is the idea behind motors and loudspeakers.

The force on the wire is:

$\vec{F}_{B} = \int I(d\vec{\ell} \times \vec{B})$

The pieces are:

• $\vec{F}_B$ is the magnetic force on the wire
• $I$ is the current through the wire
• $d\vec{\ell}$ is a small length vector along the wire, in the direction of current
• $\vec{B}$ is the external magnetic field

The direction follows the right-hand rule for cross products. For a straight wire of length $L$ in a uniform field, this simplifies to:

$F = I L B \sin\theta$

where $\theta$ is the angle between the wire and the field. So:

• Force is maximum when the wire is perpendicular to the field ($\theta = 90°$)
• Force is zero when the wire is parallel to the field ($\theta = 0°$ or $180°$)
• For other angles, the force scales with $\sin\theta$

Choosing the Right Field Formula and Justifying Direction

Problem Solving: Start by identifying the geometry. If it is a long straight wire, a loop center, a loop axis, or an arc center, you already know which derived result or symmetry argument applies. Set up $d\vec{\ell}$ and $\hat{r}$ carefully, then use symmetry to cancel components that should integrate to zero before you grind through the full integral.

Free Response: When a question asks you to justify a field or force direction, do not just write "Biot-Savart" or "right-hand rule" by name. Show the cross product direction and explain how the field is tangent to circles around the wire. Reference the right-hand rule as part of your reasoning, not as the whole answer.

For derivations, write the differential form first, state your variables and limits, and carry units through. Symbolic answers are often expected before you plug in numbers.

Common Trap: Watch the difference between $\frac{\mu_0 I}{2R}$ (center of a full loop) and $\frac{\mu_0 I}{2\pi r}$ (long straight wire). They look similar but apply to completely different geometries. For an arc that is only part of a loop, scale the full-loop result by the fraction of the circle the arc covers.

Practice Problem 1: Magnetic Field of a Straight Wire

Solution

For a long, straight wire, the magnetic field at a perpendicular distance $r$ is:

$B = \frac{\mu_0 I}{2\pi r}$

Substituting the values:

• Current $I = 5.0$ A
• Distance $r = 0.10$ m
• $\mu_0 = 4\pi \times 10^{-7}$ T·m/A

$B = \frac{4\pi \times 10^{-7} \times 5.0}{2\pi \times 0.10}$

$B = \frac{4\pi \times 10^{-7} \times 5.0}{2\pi} \times 10$

$B = 10^{-6} \times 5.0 \times 10$

$B = 5.0 \times 10^{-5}$ T, or $50$ μT

Practice Problem 2: Magnetic Field at Center of a Current Loop

Solution

The magnetic field at the center of a circular current loop is:

$B_{\text{center}} = \frac{\mu_0 I}{2R}$

Substituting the values:

• Current $I = 2.0$ A
• Radius $R = 0.050$ m
• $\mu_0 = 4\pi \times 10^{-7}$ T·m/A

$B_{\text{center}} = \frac{4\pi \times 10^{-7} \times 2.0}{2 \times 0.050}$

$B_{\text{center}} = \frac{4\pi \times 10^{-7} \times 2.0}{0.10}$

$B_{\text{center}} = 4\pi \times 10^{-7} \times 2.0 \times 10$

$B_{\text{center}} = 8\pi \times 10^{-6}$

$B_{\text{center}} = 2.51 \times 10^{-5}$ T, or about $25.1$ μT

Common Misconceptions

• Misconception: The field around a straight wire is radial.
  Actually: The field around a straight wire is not radial. It wraps in circles around the wire and is tangent to those circles, with no component pointing toward or away from the wire.
• Misconception: $B = \frac{\mu_0 I}{2R}$ works anywhere along a straight wire.
  Actually: $B = \frac{\mu_0 I}{2R}$ only works at the center of a full circular loop, not anywhere along a straight wire. Match the formula to the geometry.
• Misconception: Naming "the right-hand rule" or "Biot-Savart law" alone justifies a direction on free response.
  Actually: Naming "the right-hand rule" or "Biot-Savart law" alone does not justify a direction on free response. You need to show the actual cross product direction and reasoning.
• Misconception: The $\hat{r}$ in the Biot-Savart law points from the field point back to the wire.
  Actually: The $\hat{r}$ in the Biot-Savart law points from the current element to the field point, not from the field point back to the wire. Getting this backwards flips your direction.
• Misconception: The field from a straight wire falls off as $1/r^2$.
  Actually: More current or a smaller radius means a stronger field, but the field from a straight wire falls off as $1/r$, not $1/r^2$. Do not confuse it with electric field behavior.
• Misconception: The simplified force expression $F = ILB\sin\theta$ applies to curved wires or nonuniform fields.
  Actually: The simplified force expression $F = ILB\sin\theta$ assumes a straight wire in a uniform field. For curved wires or nonuniform fields, you have to go back to the integral $\vec{F}_{B} = \int I(d\vec{\ell} \times \vec{B})$.

Related AP Physics C: E&M Guides

• 12.2 Magnetism and Moving Charges
• 12.1 Magnetic Fields
• 12.4 Ampère's Law