AP Physics C: Electricity and Magnetism
Magnetic Forces and Fields
The Biot–Savart Law
Learning Objectives
By the end of this lesson, students should be able to:
-
State the Biot–Savart Law.
-
Explain how moving charges produce magnetic fields.
-
Calculate magnetic fields due to current elements.
-
Determine magnetic field directions using the right-hand rule.
-
Apply the Biot–Savart Law to straight wires and circular loops.
-
Understand the relationship between the Biot–Savart Law and Ampère’s Law.
Introduction
Why Do We Need the Biot–Savart Law?
Previously, we learned formulas for magnetic fields produced by:
-
Long straight wires
-
Circular loops
-
Solenoids
However, these formulas are special cases.
The Biot–Savart Law is the fundamental equation that allows us to calculate the magnetic field produced by any current distribution.
It plays a role similar to Coulomb’s Law in electrostatics.
Comparison:
Electric field from charges:
$$
\text{Coulomb’s Law}
$$
Magnetic field from currents:
$$
\text{Biot–Savart Law}
$$
Current Elements
Small Segment of Current
Consider a tiny segment of wire carrying current (I).
The segment is represented by:
$$
d\vec{\ell}
$$
called a current element.
The direction of (d\vec{\ell}) is the direction of conventional current.
Observation Point
Suppose we wish to determine the magnetic field at a point located a distance:
$$
r
$$
from the current element.
The position vector from the current element to the observation point is:
$$
\vec{r}
$$
Statement of the Biot–Savart Law
Vector Form
The magnetic field contribution from a small current element is:
$$
d\vec{B}=\frac{\mu_0}{4\pi}\frac{I,d\vec{\ell}\times\hat{r}}{r^2}
$$
where:
-
\(d\vec{B}\) = magnetic field contribution
-
\(I\) = current
-
\(d\vec{\ell}\) = current element
-
\(r\) = distance to observation point
-
\(\hat{r}\) = unit vector toward the observation point
-
\(\mu_0\) = permeability of free space
Permeability Constant
The permeability of free space is:
$$
\mu_0=4\pi\times10^{-7};T\cdot m/A
$$
This constant appears throughout magnetism.
Magnitude Form
Scalar Equation
The magnitude of the magnetic field contribution is:
$$
dB=\frac{\mu_0}{4\pi}\frac{I,d\ell,\sin\theta}{r^2}
$$
where:
-
(d\ell) = magnitude of the current element
-
(\theta) = angle between (d\vec{\ell}) and (\vec{r})
Comparison with Coulomb’s Law
Coulomb’s Law:
$$
dE=\frac{1}{4\pi\varepsilon_0}\frac{dq}{r^2}
$$
Biot–Savart Law:
$$
dB=\frac{\mu_0}{4\pi}\frac{I,d\ell,\sin\theta}{r^2}
$$
Notice the similar inverse-square dependence.
Direction of the Magnetic Field
Cross Product
The direction of:
$$
d\vec{B}
$$
is determined by:
$$
d\vec{\ell}\times\hat{r}
$$
using the right-hand rule.
Right-Hand Rule
To find the direction:
-
Point fingers along (d\vec{\ell}).
-
Curl toward (\hat{r}).
-
Thumb points in the direction of (d\vec{B}).
Integrating the Biot–Savart Law
Total Magnetic Field
A complete wire contains many current elements.
The total magnetic field is obtained by integrating:
$$
\vec{B}=\int d\vec{B}
$$
Substituting the Biot–Savart Law:
$$
\vec{B}=\frac{\mu_0 I}{4\pi}\int\frac{d\vec{\ell}\times\hat{r}}{r^2}
$$
This integral produces the magnetic field of any current distribution.
Application: Infinite Straight Wire
Geometry
Consider a very long straight wire carrying current (I).
Applying the Biot–Savart integral yields:
$$
B=\frac{\mu_0 I}{2\pi r}
$$
where:
-
(r) = perpendicular distance from the wire
Result
The magnetic field decreases as:
$$
B\propto\frac{1}{r}
$$
This is one of the most important results in magnetism.
Application: Circular Current Loop
Field at the Center
Applying the Biot–Savart Law to a circular loop gives:
$$
B=\frac{\mu_0 I}{2R}
$$
where:
-
(R) = radius of the loop
Multiple Loops
For (N) turns:
$$
B=\frac{\mu_0 N I}{2R}
$$
The magnetic field increases linearly with the number of turns.
Physical Interpretation
Dependence on Current
The magnetic field increases with current.
Doubling the current doubles the magnetic field.
$$
B\propto I
$$
Dependence on Distance
The magnetic field decreases as distance increases.
For a current element:
$$
dB\propto\frac{1}{r^2}
$$
For a complete straight wire:
$$
B\propto\frac{1}{r}
$$
The integration changes the final distance dependence.
Relationship to Ampère’s Law
Two Fundamental Methods
Magnetic fields can be found using:
-
Biot–Savart Law
-
Ampère’s Law
When to Use Each
Biot–Savart Law:
-
Works for any current distribution.
-
Usually requires integration.
Ampère’s Law:
-
Simpler.
-
Works only for highly symmetric situations.
Examples:
-
Long straight wire
-
Solenoid
-
Toroid
Example 1
Magnetic Field at the Center of a Loop
A circular loop has:
$$
I=6.0A
$$
and
$$
R=0.10m
$$
Find the magnetic field at its center.
Solution
Use:
$$
B=\frac{\mu_0 I}{2R}
$$
Substitute:
$$
B=\frac{(4\pi\times10^{-7})(6.0)}{2(0.10)}
$$
$$
B=3.77\times10^{-5}T
$$
Answer
$$
B=3.77\times10^{-5}T
$$
Example 2
Long Straight Wire
A wire carries:
$$
I=12A
$$
Find the magnetic field at:
$$
r=0.030m
$$
from the wire.
Solution
Use:
$$
B=\frac{\mu_0 I}{2\pi r}
$$
Substitute:
$$
B=\frac{(4\pi\times10^{-7})(12)}{2\pi(0.030)}
$$
$$
B=8.0\times10^{-5}T
$$
Answer
$$
B=8.0\times10^{-5}T
$$
Common AP Exam Mistakes
Mistake 1
Confusing field direction.
Always apply the right-hand rule carefully.
Remember that magnetic fields are vector quantities.
Mistake 2
Using Ampère’s Law when symmetry is absent.
Biot–Savart Law is the general approach.
Mistake 3
Forgetting the angle factor.
The Biot–Savart Law contains:
$$
\sin\theta
$$
which determines how effectively a current element contributes to the field.
AP Free-Response Strategy
Recognize Standard Geometries
Many AP questions provide situations that reduce to known results:
Straight wire:
$$
B=\frac{\mu_0 I}{2\pi r}
$$
Loop center:
$$
B=\frac{\mu_0 I}{2R}
$$
Memorizing these saves considerable time.
Identify Symmetry
Ask:
-
Is the wire straight?
-
Is the wire circular?
-
Is there cylindrical symmetry?
If so, the integration may simplify dramatically.
Use the Right-Hand Rule Early
Determine magnetic field direction before beginning calculations.
This helps avoid sign errors and earns partial credit.
Summary
Key Takeaways
-
The Biot–Savart Law is the fundamental law for magnetic fields produced by currents.
$$
d\vec{B}=\frac{\mu_0}{4\pi}\frac{I,d\vec{\ell}\times\hat{r}}{r^2}
$$
-
The magnitude form is:
$$
dB=\frac{\mu_0}{4\pi}\frac{I,d\ell,\sin\theta}{r^2}
$$
-
Magnetic field direction is determined by the right-hand rule.
-
The total magnetic field is obtained through integration:
$$
\vec{B}=\frac{\mu_0 I}{4\pi}\int\frac{d\vec{\ell}\times\hat{r}}{r^2}
$$
-
Important results derived from the Biot–Savart Law include:
Long straight wire:
$$
B=\frac{\mu_0 I}{2\pi r}
$$
Circular loop:
$$
B=\frac{\mu_0 I}{2R}
$$
-
The Biot–Savart Law is the magnetic equivalent of Coulomb’s Law and serves as one of the foundational equations of electromagnetism.