Fields of Wires and Solenoids

So far the field came from somewhere else. Now we look at the field a current makes. In 1820 Hans Christian Ørsted noticed a compass needle swing when a wire nearby carried current. A long straight wire makes field lines that are circles around it, and the field gets weaker the farther you go. You already know the force on a current; put the two together and two wires push or pull on each other.

Key idea

A long straight wire makes circles of field (thumb along $I$, fingers curl along $\vec{B}$): $$B = \frac{\mu_0 I}{2\pi r}.$$ Inside a long solenoid with $n$ turns per meter the field is nearly uniform: $$B = \mu_0 nI.$$ Two parallel wires a distance $d$ apart push or pull with $$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d},$$ and same-direction currents attract.

Here $\mu_0 = 4\pi \times 10^{-7}\ \text{T·m/A}$ (it is now measured, and it agrees with $4\pi \times 10^{-7}$ to better than a part in a billion). A handy form: $$\frac{\mu_0}{2\pi} = 2 \times 10^{-7}\ \text{T·m/A}.$$

Circles of field around a long wire. Thumb along the current, and your fingers curl the way $\vec{B}$ goes.

Why it works

Why do parallel currents attract? Wire 1 makes a field at wire 2. Use the wire rule: with both currents up the page and wire 2 to the right, wire 1's field at wire 2 points into the page. Now the force rule on wire 2: fingers up (its current), curl into the page, and the thumb points left, toward wire 1. Reverse one current and the force reverses too. The rule for charges (like repel) does not carry over, because this is a different force.

Common mistake

Expecting parallel currents to repel, the way like charges do. Check with the two right-hand rules above: same-direction currents always come out attracting. This is how the ampere used to be defined: two wires $1\ \text{m}$ apart with $1\ \text{A}$ each pull with $2 \times 10^{-7}\ \text{N}$ per meter.

Worked example: A wire and a compass

A long wire carries $10\ \text{A}$. Find the field $5.0\ \text{cm}$ away.

$$\begin{aligned} B &= \frac{\mu_0 I}{2\pi r} = (2 \times 10^{-7})\frac{10}{0.050} \\ &= 4.0 \times 10^{-5}\ \text{T}. \end{aligned}$$
That is $40\ \mu\text{T}$, close to Earth's own field, which is why a compass near a wire swings so much.

Answer: $4.0 \times 10^{-5}\ \text{T}$

Worked example: Inside a solenoid

A solenoid $0.25\ \text{m}$ long has $500$ turns and carries $2.0\ \text{A}$. Find the field inside.

Turns per meter: $n = 500/0.25\ \text{m} = 2000\ \text{m}^{-1}$.
$$\begin{aligned} B &= \mu_0 nI = (4\pi \times 10^{-7})(2000)(2.0) \\ &= 5.0 \times 10^{-3}\ \text{T}. \end{aligned}$$
The field depends on turns per meter, not on the solenoid's width.

Answer: $5.0\ \text{mT}$

Worked example: Two wires

Two long parallel wires $1.0\ \text{cm}$ apart carry $5.0\ \text{A}$ and $8.0\ \text{A}$ in the same direction. Find the force per meter.

$$\begin{aligned} \frac{F}{L} &= \frac{\mu_0 I_1 I_2}{2\pi d} = (2 \times 10^{-7})\frac{(5.0)(8.0)}{0.010} \\ &= 8.0 \times 10^{-4}\ \text{N/m}. \end{aligned}$$
Same direction, so they attract. Each wire feels the same size force (Newton's third law), even though the currents differ.

Answer: $8.0 \times 10^{-4}\ \text{N/m}$, attracting

Fields of Wires and Solenoids

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