Magnetostatics: Biot-Savart & Ampère's Law
Magnetostatics: Biot-Savart & Ampère's Law
Magnetostatics describes magnetic fields produced by steady currents. The Biot-Savart law gives the field element \(d\mathbf{B}\) from a current element, while Ampère's law provides a powerful symmetry-based tool.
Definition
The magnetic field from a current element is \(d\mathbf{B} = \frac{\mu_0 I}{4\pi}\frac{d\mathbf{l}\times\hat{r}}{r^2}\). For a long straight wire: \(B = \mu_0 I/(2\pi r)\).
Key Result
Ampère's law (static): \(\oint \mathbf{B}\cdot d\mathbf{l} = \mu_0 I_{enc}\). Inside a solenoid of \(n\) turns/m: \(B = \mu_0 n I\).
Example 1
A toroidal solenoid with \(N\) turns, mean radius \(R\): \(B = \mu_0 N I/(2\pi R)\) inside the torus, essentially zero outside.
Example 2
Two parallel wires carrying currents \(I_1, I_2\) separated by \(d\) attract with force per unit length \(F/L = \mu_0 I_1 I_2/(2\pi d)\).
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Practice
- Apply Ampère's law to find B inside an infinite solenoid.
- Compute the torque on a magnetic dipole in a uniform field.
- What is the Lorentz force on a charge in combined E and B fields?
- Explain why magnetic monopoles have never been observed.
Show Answer Key
1. Amperian loop of length $l$ inside solenoid ($n$ turns/length, current $I$): $\oint\mathbf{B}\cdot d\mathbf{l} = Bl = \mu_0 nIl$. So $B = \mu_0 nI$ (uniform inside). Outside: $B = 0$ (field lines close at infinity for ideal infinite solenoid).
2. Torque: $\boldsymbol{\tau} = \mathbf{m}\times\mathbf{B}$ where $\mathbf{m} = NIA\hat{n}$ is the magnetic dipole moment. $|\tau| = mB\sin\theta$ (maximum when $\mathbf{m}\perp\mathbf{B}$, zero when aligned). Energy: $U = -\mathbf{m}\cdot\mathbf{B}$.
3. Lorentz force: $\mathbf{F} = q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$. The electric force $q\mathbf{E}$ acts along $\mathbf{E}$ regardless of velocity. The magnetic force $q\mathbf{v}\times\mathbf{B}$ is perpendicular to both $\mathbf{v}$ and $\mathbf{B}$, doing no work (only changes direction).
4. Maxwell's equations are symmetric between $\mathbf{E}$ and $\mathbf{B}$ except for $\nabla\cdot\mathbf{B}=0$ (no magnetic charges). If monopoles existed, we'd have $\nabla\cdot\mathbf{B}=\mu_0\rho_m$ and a magnetic current term. Despite theoretical appeal (Dirac quantization, GUT predictions), no experimental evidence exists. Searches at the LHC, in cosmic rays, and in condensed matter analogs continue.