Calculating Magnetic Fields with Biot-Savart and Ampère's Laws

1. The Biot-Savart Law: The Magnetic Analogue of Coulomb's Law

Just as Coulomb's Law allows us to calculate the electric field from a distribution of charges, the Biot-Savart Law allows us to calculate the magnetic field from a distribution of currents. It states that an infinitesimal segment of current-carrying wire, $Id\mathbf{l}$, produces an infinitesimal magnetic field, $d\mathbf{B}$, at a position $\mathbf{r}$ away.

The law is expressed as:
$d\mathbf{B}=\frac{{\mu }_0}{4\pi }\frac{Id\mathbf{l}\times \hat{\mathbf{r}}}{r^2}$
To find the total magnetic field $\mathbf{B}$ at a point, we must integrate this expression over the entire current distribution:
$\mathbf{B}=\frac{{\mu }_{0}I}{4\pi }\int \frac{d\mathbf{l}\times \hat{\mathbf{r}}}{r^2}$

Let's break down the components:

• ${\mu }_0$ is the permeability of free space, a fundamental constant with the value $4\pi \times 10^{-7}\text{T}\cdot \text{m}/\text{A}$.
• $d\mathbf{l}$ is an infinitesimal vector pointing in the direction of the current $I$.
• $\mathbf{r}$ is the position vector from the current element $d\mathbf{l}$ to the point where the field is being calculated, and $r=|\mathbf{r}|$.
• $\hat{\mathbf{r}}$ is the unit vector $\mathbf{r}/r$.
• The cross product $d\mathbf{l}\times \hat{\mathbf{r}}$ is crucial. It means the resulting magnetic field $d\mathbf{B}$ is perpendicular to both the current direction and the vector connecting the current to the point of interest. The direction is given by the right-hand rule.

Applying the Biot-Savart Law: Field of a Straight Wire

The canonical application of the Biot-Savart Law is calculating the magnetic field produced by a long, straight wire. This requires setting up and solving a vector integral, a fantastic exercise for your calculus skills.

The video below provides a detailed, step-by-step derivation. It starts by finding the field from a finite rod and then takes the limit as the rod becomes infinitely long. This approach is more instructive than tackling the infinite case directly.

As you saw in the video, the integration yields a simple and famous result for the magnetic field at a radial distance $r$ from an infinitely long, straight wire carrying current $I$:
$\mathbf{B}=\frac{{\mu }_{0}I}{2\pi r}\hat{\mathbf{\varphi }}$
Here, $\hat{\mathbf{\varphi }}$ is the unit vector that curls around the wire according to the right-hand rule.

Here is a static image summarizing the derivation for your reference.

2. Ampère's Law: The Magnetic Analogue of Gauss's Law

While the Biot-Savart Law is always true, the integral can be very difficult to solve for complex geometries. For situations with high symmetry, we have a much more powerful tool: Ampère's Law.

In electrostatics, Gauss's Law relates the flux of the electric field through a closed surface to the enclosed charge. Ampère's Law is similar in spirit but relates the circulation of the magnetic field around a closed loop to the total current passing through that loop.

In its integral form, Ampère's Law is:
${\oint }_C\mathbf{B}\cdot d\mathbf{l}={\mu }_0{I}_{\text{enc}}$
Let's break this down:

• The integral ${\oint }_C$ is a line integral around a closed path, or loop, called an "Amperian loop".
• $\mathbf{B}\cdot d\mathbf{l}$ measures how much the magnetic field aligns with the path at each point.
• $I_{\text{enc}}$ is the net electric current that passes through the surface bounded by the loop $C$. The direction of "passing through" is determined by a right-hand rule applied to the direction of the loop integration.

When can we use Ampère's Law?
Just like Gauss's Law, Ampère's Law is most useful when symmetry allows us to simplify the line integral. We need to be able to choose an Amperian loop where:

1. $\mathbf{B}$ is tangent to the loop and has a constant magnitude $B$. The integral becomes $\oint Bdl=B\oint dl=B\cdot (\text{length of loop})$.
2. $\mathbf{B}$ is perpendicular to the loop. The integral becomes $\oint \mathbf{B}\cdot d\mathbf{l}=0$.

Let's see this in action.

Application 1: Infinite Straight Wire (Revisited)

The high degree of cylindrical symmetry makes the infinite wire a perfect candidate for Ampère's Law. The following reading walks through this classic example.

As you can see, by choosing a circular Amperian loop of radius $r$ centered on the wire, the line integral simplifies beautifully:
$\oint \mathbf{B}\cdot d\mathbf{l}=\oint B(r)\hat{\mathbf{\varphi }}\cdot dl\hat{\mathbf{\varphi }}=B(r)\oint dl=B(r)\cdot (2\pi r)$
Since $I_{\text{enc}}=I$, Ampère's law gives $B(r)\cdot (2\pi r)={\mu }_{0}I$, which immediately recovers the result we found with the Biot-Savart Law: $B=\frac{{\mu }_{0}I}{2\pi r}$. The difference in effort is remarkable.

Application 2: The Ideal Solenoid

Another crucial application is finding the magnetic field inside a solenoid—a coil of wire wound into a tight helix. For an infinitely long solenoid, the field inside is remarkably uniform, and the field outside is zero.

The following text provides a clear, symmetry-based derivation of this result using a cleverly chosen rectangular Amperian loop.

This derivation shows that the magnetic field inside a long solenoid with $n$ turns per unit length carrying current $I$ is:
$\mathbf{B}={\mu }_{0}nI\hat{\mathbf{z}}$
And the field outside is $\mathbf{B}=0$. This ability to create a strong, uniform magnetic field makes solenoids a vital component in countless engineering and scientific applications, from electromagnets and inductors to MRI machines.

3. The Differential Form of Ampère's Law

The integral form of Ampère's Law relates the magnetic field over a macroscopic loop to the total current. Physics often seeks a local relationship—one that holds at every single point in space. This is given by the differential form of the law.

To get there, we introduce current density, $\mathbf{J}$, which is the vector representing the current per unit area at a point. The total current $I_{\text{enc}}$ is the flux of the current density through the surface $S$: $I_{\text{enc}}={\int }_S\mathbf{J}\cdot d\mathbf{A}$.

By applying Stokes' Theorem from vector calculus, which states ${\oint }_C\mathbf{F}\cdot d\mathbf{l}={\int }_S(\nabla \times \mathbf{F})\cdot d\mathbf{A}$, we can transform the line integral of $\mathbf{B}$ into a surface integral of its curl, $\nabla \times \mathbf{B}$.

The following video masterfully derives this relationship. For our current topic of magnetostatics, the electric field is constant, so the term involving $\partial E/\partial t$ is zero.

By equating the integrands, we arrive at the differential form of Ampère's Law for magnetostatics:
$\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$
This beautiful equation is a local statement: the amount the magnetic field "curls" or circulates at a point is directly proportional to the current density at that same point. This is one of the pillars of classical electromagnetism.

Conclusion

In this lesson, we have equipped ourselves with the two foundational laws of magnetostatics for calculating magnetic fields from steady currents.

Key Takeaways:

• The Biot-Savart Law ($\mathbf{B}=\frac{{\mu }_{0}I}{4\pi }\int \frac{d\mathbf{l}\times \hat{\mathbf{r}}}{r^2}$) is a general, fundamental law that works for any current geometry but often requires complex integration.
• Ampère's Law ($\oint \mathbf{B}\cdot d\mathbf{l}={\mu }_0{I}_{\text{enc}}$) is a powerful shortcut for calculating magnetic fields in situations with high cylindrical, toroidal, or planar symmetry.
• We derived the magnetic fields for two canonical systems: the long straight wire ($B=\frac{{\mu }_{0}I}{2\pi r}$) and the ideal solenoid ($B={\mu }_{0}nI$).
• The differential form of Ampère's Law, $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$, provides a local relationship between the curl of the magnetic field and the current density at a point. It is derived from the integral form using Stokes' Theorem.

Preview of the Next Lesson:
Our exploration of magnetostatics has just begun. While Ampère's law is elegant, its reliance on symmetry is limiting. In the next lesson, we will introduce the magnetic vector potential, $\mathbf{A}$. Similar to how the scalar potential $\varphi$ simplified many electrostatics problems, the vector potential provides a more advanced and often more powerful framework for finding the magnetic field, especially for complex systems. We will define it through the relation $\mathbf{B}=\nabla \times \mathbf{A}$ and see how it connects back to the Biot-Savart law.