In our study of electricity, we made extensive use of the idea of a uniform electric field: a field that is the same at every point in space. We found that two closely spaced, parallel charged plates generate a uniform electric field between them, and this was one reason we focused so much attention on the parallel-plate capacitor.

Similarly, there are many applications of magnetism for which we would like to generate a uniform magnetic field, a field having the same magnitude and the same direction at every point within some region of space. Neither a long, straight wire nor a current loop produces a uniform magnetic field.

In practice, a uniform magnetic field is generated with a solenoid. A solenoid, shown in FIGURE 29.28, is a helical coil of wire of length L that is formed with N loops—usually called turns—of wire. The same current I passes through each turn as it travels through the coil. Solenoids may have hundreds or thousands of turns, sometimes wrapped in several layers.

FIGURE 29.28 A solenoid.

We can understand a solenoid by thinking of it as a stack of current loops. FIGURE 29.29a shows the magnetic field of a single current loop at three points on the axis and three points equally distant from the axis. The field directly above the loop is opposite in direction to the field inside the loop. FIGURE 29.29b then shows three parallel loops. We can use information from Figure 29.29b to draw the magnetic fields of each loop at the center of loop 2 and at a point above loop 2.

FIGURE 29.29 Using superposition to find the magnetic field of a stack of current loops.

Full Alternative Text

The superposition of the three fields at the center of loop 2 produces a stronger field than that of loop 2 alone. But the superposition at the point above loop 2 produces a net magnetic field that is very much weaker than the field at the center of the loop. We’ve used only three current loops to illustrate the idea, but these tendencies are reinforced by including more loops. With many current loops along the same axis, the field in the center is strong and roughly parallel to the axis, whereas the field outside the loops is very close to zero.

FIGURE 29.30a is a photo of the magnetic field of a short solenoid. You can see that the magnetic field inside the coils is nearly uniform (i.e., the field lines are nearly parallel) and the field outside is much weaker. Our goal of producing a uniform magnetic field can be achieved by increasing the number of coils until we have an ideal solenoid that is infinitely long and in which the coils are as close together as possible. As FIGURE 29.30b shows, the magnetic field inside an ideal solenoid is uniform and parallel to the axis; the magnetic field outside is zero. No real solenoid is ideal, but a very uniform magnetic field can be produced near the center of a tightly wound solenoid whose length is much larger than its diameter.

FIGURE 29.30 The magnetic field of a solenoid.

We can use Ampère’s law to calculate the field of an ideal solenoid. FIGURE 29.31 shows a cross section through an infinitely long solenoid. The integration path that we’ll use is a rectangle of width l, enclosing N turns of the solenoid coil. Because this is a mathematical curve, not a physical boundary, there’s no difficulty with letting it protrude through the wall of the solenoid wherever we wish. The solenoid’s magnetic field direction, given by the right-hand rule, is left to right, so we’ll integrate around this path in the ccw direction.

FIGURE 29.31 A closed path inside and outside an ideal solenoid.

Each of the N wires enclosed by the integration path carries current I, so the total current passing through the rectangle is $I_{\text{through}}=NI$. Ampère’s law is thus

The line integral around this path is the sum of the line integrals along each side. Along the bottom, where $\vec{B}$ is parallel to $d\vec{s}$ and of constant value B, the integral is simply Bl. The integral along the top is zero because the magnetic field outside an ideal solenoid is zero.

The left and right sides sample the magnetic field both inside and outside the solenoid. The magnetic field outside is zero, and the interior magnetic field is everywhere perpendicular to the line of integration. Consequently, as we recognized in Option 1 of Tactics Box 29.3, the line integral is zero.

Only the integral along the bottom path is nonzero, leading to

Thus the strength of the uniform magnetic field inside a solenoid is

where $n=N\text{/}l$ is the number of turns per unit length. Measurements that need a uniform magnetic field are often conducted inside a solenoid, which can be built quite large.

EXAMPLE 29.9 Generating an MRI magnetic field

A 1.0-m-long MRI solenoid generates a 1.2 T magnetic field. To produce such a large field, the solenoid is wrapped with superconducting wire that can carry a 100 A current. How many turns of wire does the solenoid need?

MODEL Assume that the solenoid is ideal.

SOLVE Generating a magnetic field with a solenoid is a trade-off between current and turns of wire. A larger current requires fewer turns, but the resistance of ordinary wires causes them to overheat if the current is too large. For a superconducting wire that can carry 100 A with no resistance, we can use Equation 29.17 to find the required number of turns:

$$N={\displaystyle \frac{lB}{{\mu }_{0}I}}={\displaystyle \frac{(1.0\text{m)(1.2 T})}{(1.26\times {10}^{-6}\text{T m/A)(100 A})}}=9500\text{turns}$$

REVIEW The solenoid coil requires a large number of turns, but that’s not surprising for generating a very strong field. If the wires are 1 mm in diameter, there would be 10 layers with approximately 1000 turns per layer.

The magnetic field of a finite-length solenoid is approximately uniform inside the solenoid and weak, but not zero, outside. As FIGURE 29.32 shows, the magnetic field outside the solenoid looks like that of a bar magnet. Thus a solenoid is an electromagnet, and you can use the right-hand rule to identify the north-pole end. A solenoid with many turns and a large current can be a very powerful magnet.

FIGURE 29.32 The magnetic fields of a finite-length solenoid and of a bar magnet.