Moving Charges and MagnetismMagnetic Force
Before we meet the magnetic field, recall how Chapter 1 described the electric field. A charge Q, the source, sets up a field all around it,
where is the unit vector along r. A charge q placed in that field feels a force
The field is more than bookkeeping: it carries energy and momentum and takes time to spread out. In this chapter every field stays constant in time, and the fields of several sources add as vectors: the principle of superposition.
Moving charges and currents set up a second vector field, the magnetic field , as well as any electric field. It also obeys superposition: the field of several sources is the vector sum of their separate fields. Which of these produce a magnetic field?
Charges at rest make only an electric field. Moving charges, currents and magnets make a magnetic field as well. In a magnet the sources are tiny circulating currents and the built-in magnetic moments of electrons, which you will meet in the next chapter.
Magnetic Field, Lorentz Force
A point charge q moves with velocity v through a region with an electric field E and a magnetic field B. The total force on it is
This is the Lorentz force, named after the Dutch physicist Hendrik Lorentz, who wrote it down from the experiments of Ampere and others. The magnetic part, , is a cross product. Its size is
where θ is the angle between v and B, and it points at right angles to both. Here the field points to the right, across the page. Drag the tip of the velocity round and watch the force.
The green graph is sin θ. Three features of the magnetic force stand out. (i) It depends on q, v and B, and a negative charge feels the opposite force to a positive one. (ii) It vanishes when v is parallel or antiparallel to B; otherwise it acts sideways, at right angles to both v and B. (iii) A charge at rest, with v = 0, feels no magnetic force at all. Only a moving charge feels a magnetic force.
Tip
To find the direction of , point the fingers of your right hand along v and curl them towards B through the smaller angle. Your thumb gives the force on a positive charge. For an electron, reverse it.
How Big Is a Tesla?
The force law also defines the unit of B. Send a charge of 1 C at 1 m/s at right angles to the field: if the force on it is 1 N, the field is one tesla (T), named after Nikola Tesla (1856–1943). From ,
A tesla is a large field. A smaller, non-SI unit, the gauss, is often used: 1 G = 10⁻⁴ T. The Earth's field is about 3.6 × 10⁻⁵ T, which is
Arrange these from the weakest field to the strongest.
The Earth gives a few times 10⁻⁵ T, a fridge magnet a few thousandths of a tesla, an MRI scanner 1.5 to 3 T, and a neutron star 10⁸ T or more.
Magnetic Force on a Current-carrying Conductor
The force on one moving charge adds up to a force on a whole wire. Take a straight rod of length l and cross-section A, with n free electrons per unit volume, so it holds nlA of them. In a steady current I each one drifts with the average drift velocity of Chapter 3. In an external field B the total force on them is
The electrons drift to the left, so the current flows to the right. With B into the page each electron is pushed up, and so the whole rod feels the force F. Now is the current density j, and , so
where has the length of the rod and points along the current. The current I is not a vector: the vector sign has moved from j to l. B here is the external field, not the field the rod makes itself. For a wire of any shape, add up the forces on short straight pieces:
A closed loop of wire, of any shape, carries a steady current in a uniform field. The net force on the loop is
In a uniform field, , and the little steps dl round a closed loop add up to zero. So a current loop feels no net force in a uniform field. It can still feel a torque, and that is what turns motors and meters (Section 4.9).
Example 4.1: Hold a wire up with a magnetic field. A straight wire of mass 200 g and length 1.5 m carries a current of 2 A. A uniform horizontal magnetic field at right angles to the wire holds it up in mid-air (Fig. 4.3). How strong is the field?
The upward force IlB must balance the weight mg. Set the field: B =
At balance the two forces are equal:
Only the mass per unit length, m/l, really matters. The Earth's field, about 4 × 10⁻⁵ T, is far too small to notice here, so we have ignored it.
Example 4.2: Which way is the push? The magnetic field points along the positive y-axis, and a charged particle moves along the positive x-axis with the velocity shown (Fig. 4.4). Which way is the Lorentz force on (a) an electron, (b) a proton?
points along , the +z axis. So the force on (a) the electron, a negative charge, points along
The two forces point opposite ways along the z-axis: the electron's negative charge reverses the direction of .
Speed Round: Which Way Is the Force?
, , . Swapping the order flips the sign, and a negative charge flips it again. On the page, +x is to the right, +y up the page and +z out of the page. Get five right to move on.