Chapter 32 – Magnetic Fields

This chapter has a lot of material. The chapter begins with an overview of magnetism. The key points are the following: magnets apparently only come in North Pole – South Pole pairs, that is dipoles, magnetic fields are caused by moving charges, and moving charges in a magnetic field feel a force which depends on how fast the charge is moving. Note that “natural” magnets get there inherent magnetism from atomic-sized moving charges.

1. Currents are formed when charges move around. Consequently the magnetic field set up by a moving charge is analogous to the magnetic field set up by a current carrying wire. The “right hand rule” is used to determine the vector sense of a magnetic field produced by moving charges or currents. Imagine grabbing a current carrying wire with your right hand while your thumb points in the direction of the current. The magnetic field then curls around the wire in the same direction that your fingers curl around the wire. For example, for a wire perpendicular to this page with current flowing into the paper, the magnetic field points clockwise around the wire. The right hand rule lets you get a qualitative sense of the magnetic field produced by a current or a moving charge. Being about to correctly use the right hand rule is very important.

2. The Biot-Savart Law is the analog to E = q r/(4 πε₀ r²) where r is a unit vector pointing from the charge q producing the electric field to the point at which the electric field is being found. The Biot-Savart Law for the magnetic field produced by a point charge q moving with velocity v is more difficult to apply because it involves the vector cross product (a mathematical representation of the right hand rule) but is structurally the same as the equation for E,

B = μ₀ q v x r/(4 π r²).

The constant μ₀ is the permeability and equals 4 π x 10^-7 Tesla.meter/Amp. Iimagine a small chunk of charge, Δq, in a wire moving a distance Δs along the wire in time Δt. Then the velocity of Δq is just Δs/Δt . Then we can replace qv in the Biot-Savart Law by Δq Δs/Δt. Move the Δt from below the Δs to below the Δq, remember I = Δq/ Δt, to get I Δs instead of q v.

ΔB = μ₀ I Δs x r/(4 π r²),

which gives the little bit of magnetic field due to a little chunk of current carrying wire Δs. To get the net field, add together all the little contributions, that is integrate over the length of wire producing the magnetic field. This method is used in the book to find the field due to a long straight wire and a circular loop of wire.

3. The Biot-Savart Law can be used to find the magnetic field produced by an infinite wire. For a wire perpendicular to this page and current into the page, the cross product in the Biot-Savart Law gives a magnetic field that is tangent to circles centered on the wire with B pointing in a clockwise direction. The magnitude of B is μ₀ I /(2 π r), where r is the distance to the wire.

4. Analogously, the Biot-Savart Law can be used to find the magnetic field on the axis of symmetry of a circular current carrying loop. Let the radius of the loop be R, the current I, and the distance to the plane of the loop z. Then the magnetic field at z has magnitude μ₀ I R²/2(R² + z²)^3/2. At the center of the loop where z = 0 B is just μ₀ I/2R.

5. Far away from the loop, z >> R, where the R² in the denominator can be ignored compared to the z², the magnetic field becomes, B = μ₀ I (πR²)/(2 π z³). This looks very much like the electric field of a dipole along a line parallel to p the dipole moment,

E dipole = 2 p/(4 πε₀ z³) B far from a current carrying loop =2 μ μ₀/(4 πz³), where μ = IA, the current in the loop times the area of the loop, A = πR². The direction of the magnetic dipole is the same as B, the magnetic field at the center of the loop.

6. Ampere’s Law is an alternative way of finding the magnetic field in situations that have a lot of symmetry.

∫BAt ds = μ₀ I, the integral is around a closed loop and I is the total current passing inside the loop.

Ampere’s Law applies to a loop enclosing a current I. The path integral sums the contributions of the vector dot product of the magnetic field at a point on the loop with t, the unit vector tangent to the curve at that point on the loop. Ampere’s Law, like Gauss’ Law, is very useful in situations with lots of symmetry.

7. A solenoid consists of a hollow cylinder with many coils of wire wrapped around it. An ideal solenoid produces zero magnetic field outside the cylinder and a uniform field inside. It is the analog of the ideal parallel plate capacitor which produced a constant E field between its plates. B inside the solenoid = I μ₀ n, where n = N/L, loops of wire/meter. The direction of the field inside is found by using the right-hand rule.

8. The magnetic field B exerts a force on a moving charge given by, F = q v x B. If instead of a point charge, the magnetic field acts on a short section of current carrying wire, ds, dF = I ds x B where ds points in the direction of the current. For two parallel wires carrying currents I₁ and I₂ separated by a distance d, the force/length is given by,

F/length = μ₀ I₁ I₂/2 π d. The force is attractive when the currents are in the same direction and repulsive when the currents are in the opposite direction.

(Note that the force on an infinite wire due to another infinite wire is infinite! That is why the above equation is for force/length.)

9. As noted above, a current loop acts like a magnetic dipole. The torque exerted on a magnetic dipole by a constant magnetic field is μ x B which is analogous to the torque on an electric dipole, p x E. Most electric motors operate using a current to produce a magnetic field which then acts on a current carrying loop of wire causing it to rotate.

10. The table below summarizes important analogies between electric and magnetic fields.

Electric Field	Magnetic Field
Charges, moving or stationary, create electric fields.	Moving charges (currents) create magnetic fields.
Elementary unit of charge is the point charge.	Elementary unit of magnetism is the dipole with a north and south pole. No one has ever seen an isolated north or south magnetic pole.
Electric field produces a force on a point charge, F = q E.	Magnetic field produces a force on a moving charge, F = q vxB. There is no magnetic force on stationary charges. Also the above equation is intrinsically three dimensional in that the force vector is perpendicular to the plane formed by v and B.
Coulomb’s Law: E = q r/(4 πε0 r^2) where r is a unit vector pointing from the charge q producing the electric field to the point at which the electric field is being found.	Biot-Savart Law: B = μ0 q v x r/(4 π r^2) where v is the velocity of the point charge producing the magnetic field. r is defined here as it was for Coulomb’s Law.
Gauss’ Law: ∫EAn ds = q/ε0 Gauss’s Law applies to a closed surface and q is the total charge inside the surface. The surface integral is the dot product of the electric field with the outward pointing normal vector, n, summed over the closed surface.	Ampere’s Law: ∫BAt ds = μ0 I Ampere’s Law applies to a loop enclosing the current I. The path integral is the magnetic field dotted into the normalized tangent vector, t, to the curve forming the loop summed over the closed loop.
An electric field produces a torque on an electric dipole p, torque = p x E.	A magnetic field produces a torque on a magnetic dipole, μ, torque = μ x B