Hearing all we do about Einstein, we sometimes get the impression that he invented relativity out of nothing. On the contrary, one of Einstein’s motivations was to solve difficulties in knowing how different observers see magnetic and electric fields.
MAGNETIC FIELD PRODUCED BY A CURRENT-CARRYING CIRCULAR LOOP
The magnetic field near a current-carrying loop of wire is shown in Figure 3. Both the direction and the magnitude of the magnetic field produced by a current-carrying loop are complex. RHR-2 can be used to give the direction of the field near the loop, but mapping with compasses and the rules about field lines given in Magnetic Fields and Magnetic Field Lines are needed for more detail. There is a simple formula for the magnetic field strength at the center of a circular loop. It is
\(B = \frac{\mu _{0} I}{2R} (\mathrm{at \thinspace center \thinspace of \thinspace loop}),\)
where \(R\) is the radius of the loop. This equation is very similar to that for a straight wire, but it is valid only at the center of a circular loop of wire. The similarity of the equations does indicate that similar field strength can be obtained at the center of a loop. One way to get a larger field is to have \(N\) loops; then, the field is \(B = N_{\mu_{0}} I /(2R)\). Note that the larger the loop, the smaller the field at its center, because the current is farther away.
Figure 3. (a) RHR-2 gives the direction of the magnetic field inside and outside a current-carrying loop. (b) More detailed mapping with compasses or with a Hall probe completes the picture. The field is similar to that of a bar magnet.
MAGNETIC FIELD PRODUCED BY A CURRENT-CARRYING SOLENOID
A solenoid is a long coil of wire (with many turns or loops, as opposed to a flat loop). Because of its shape, the field inside a solenoid can be very uniform, and also very strong. The field just outside the coils is nearly zero. Figure 4 shows how the field looks and how its direction is given by RHR-2.
Figure 4. (a) Because of its shape, the field inside a solenoid of length \(l\) is remarkably uniform in magnitude and direction, as indicated by the straight and uniformly spaced field lines. The field outside the coils is nearly zero. (b) This cutaway shows the magnetic field generated by the current in the solenoid.
The magnetic field inside of a current-carrying solenoid is very uniform in direction and magnitude. Only near the ends does it begin to weaken and change direction. The field outside has similar complexities to flat loops and bar magnets, but the magnetic field strength inside a solenoid is simply
\(B = \mu _{0}nI (\mathrm{inside \thinspace a \thinspace solenoid}),\)
where \(n\) is the number of loops per unit length of the solenoid (\(n = N/l\), with \(N\) being the number of loops and \(l\) the length). Note that \(B\) is the field strength anywhere in the uniform region of the interior and not just at the center. Large uniform fields spread over a large volume are possible with solenoids, as Example #2 implies.
EXAMPLE #2
What is the field inside a 2.00-m-long solenoid that has 2000 loops and carries a 1600-A current?
Strategy
To find the field strength inside a solenoid, we use \(B = \mu _{0}nI\). First, we note the number of loops per unit length is
\(n = \frac{N}{l} = \frac{2000}{2.00 \thinspace \mathrm{m^{-1}}} = 10 \thinspace \mathrm{cm^{-1}}.\)
Solution
Substituting known values gives
\(B = \mu _{0}nI = (4 \pi \times 10^{-7} \thinspace \mathrm{T \cdot m/A})(1000 \thinspace \mathrm{m^{-1}})(1600 \thinspace \mathrm{A})\)
\(= 2.01 \thinspace \mathrm{T}.\)
Discussion
This is a large field strength that could be established over a large-diameter solenoid, such as in medical uses of magnetic resonance imaging (MRI). The very large current is an indication that the fields of this strength are not easily achieved, however. Such a large current through 1000 loops squeezed into a meter’s length would produce significant heating. Higher currents can be achieved by using superconducting wires, although this is expensive. There is an upper limit to the current, since the superconducting state is disrupted by very large magnetic fields.
APPLYING THE SCIENCE PRACTICES: CHARGED PARTICLE IN A MAGNETIC FIELD
There are interesting variations of the flat coil and solenoid. For example, the toroidal coil used to confine the reactive particles in tokamaks is much like a solenoid bent into a circle. The field inside a toroid is very strong but circular. Charged particles travel in circles, following the field lines, and collide with one another, perhaps inducing fusion. But the charged particles do not cross field lines and escape the toroid. A whole range of coil shapes are used to produce all sorts of magnetic field shapes. Adding ferromagnetic materials produces greater field strengths and can have a significant effect on the shape of the field. Ferromagnetic materials tend to trap magnetic fields (the field lines bend into the ferromagnetic material, leaving weaker fields outside it) and are used as shields for devices that are adversely affected by magnetic fields, including the Earth’s magnetic field.
PHET EXPLORATIONS: GENERATOR
