Biot-Savart and Ampere's Circuital Law: Quick Revision

A brief note on Biot-Savart and Ampere's Circuital Law. Read it before a test. Try each question at the end before you open its answer.

The Biot-Savart law

• A piece of current: $dB = \dfrac{\mu_0}{4\pi}\dfrac{I\,dl\sin\theta}{r^2}$, with $\dfrac{\mu_0}{4\pi} = 10^{-7}\ \text{T m/A}$. The field is along $d\vec{l}\times\vec{r}$.
• Direction: right-hand rule. Thumb along the current, fingers along the field.
• Wire of any length: $B = \dfrac{\mu_0 I}{4\pi d}(\sin\varphi_1 + \sin\varphi_2)$.
• Very long wire: $B = \dfrac{\mu_0 I}{2\pi d}$.
• Arc, at its centre: $B = \dfrac{\mu_0 I\theta}{4\pi R}$. Full loop of $N$ turns: $B = \dfrac{\mu_0 N I}{2R}$.
• Axis of a loop: $B = \dfrac{\mu_0 N I R^2}{2(R^2 + x^2)^{3/2}}$.

Ampere's circuital law

• The law: $\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{enc}$. Only the net current through the loop counts.
• Long wire: $B = \dfrac{\mu_0 I}{2\pi r}$.
• Inside a round wire with an even current: $B = \dfrac{\mu_0 I\,r}{2\pi a^2}$. It is zero on the axis and largest at the surface.
• Inside a hollow pipe: $B = 0$.
• Zero circulation does not mean zero field.

Solenoid, toroid and parallel wires

• Long solenoid: $B = \mu_0 n I$ inside, with $n = N/L$. At an end the field is half as much. Outside it is almost zero.
• Toroid: $B = \dfrac{\mu_0 N I}{2\pi r}$ inside the ring. In the hole and outside it, $B = 0$.
• Parallel wires: $\dfrac{F}{L} = \dfrac{\mu_0 I_1 I_2}{2\pi d}$ on each metre. Same direction attracts. Opposite direction repels.

A worked example

A solenoid has $1000$ turns on a tube $25$ cm long and carries $0.60$ A. Find the field inside.

1. Turns in each metre. $n = \dfrac{1000}{0.25\ \text{m}} = 4000\ \text{m}^{-1}$.
2. The field. $B = \mu_0 n I = 4\pi \times 10^{-7}\ \text{T m/A} \times 4000\ \text{m}^{-1} \times 0.60\ \text{A} = 3.0 \times 10^{-3}\ \text{T}$, that is $3.0$ mT.

Practise with new numbers

Check yourself

1. The distance from a long straight wire is made three times larger, with the same current. What happens to the field?

2. Two long parallel wires carry currents in the same direction. Do they attract or repel?

3. A toroid carries a current. What is the field in its hole?