Problem 5.11 Find the magnetic field at point P on the axis of a tightly wound solenoid (helical coil) consisting of n turns per unit length wrapped around a cylin- drical tube of radius a and carrying current I (Fig. 5.25). Express your answer in terms of 01 and 02 (it's easiest that way). Consider the turns to be essentially circu- lar, and use the result of Ex. 5.6. What is the field on the axis of an infinite solenoid (infinite in both directions)? a FIGURE 5.25 Ꮎ, P Ꮎ, Solenoids and Helmholtz coils: 1. Griffiths 5.11. This is a finite solenoid problem that we already did in lecture, so you can reproduce the integration, but note that the angles defined in Fig. 5.24 in the textbook are different from the ones that we used in lecture. In addition to deriving the result (B field on axis everywhere), plot the B field for a solenoid of radius R and length L = 10 R. How close does the field at the center come to the field of an infinite solenoid μon? [Hint: we know the magnetic field of a ring of radius R carrying a current I on axis is given by B = µoIR²/[2(R² + z²)³/2], so breaking the solenoid into a bunch of rings should do the trick. The plot of B vs. position in the solenoid should look something like what I show below, but of course you will do a much better job: -5R Mon I 5R 2 = R is absent. 2. Now imagine a very, very long solenoid, but with a gap: a piece of length L What is the magnetic field right at the center of the gap? We will soon see that gapped solenoids are relevant to some practical problems. [Hint: think of superposition, the result of problem #1 might come in handy].

Problem 5.11 Find the magnetic field at point P on the axis of a tightly wound solenoid (helical coil) consisting of n turns per unit length wrapped around a cylin- drical tube of radius a and carrying current I (Fig. 5.25). Express your answer in terms of 01 and 02 (it's easiest that way). Consider the turns to be essentially circu- lar, and use the result of Ex. 5.6. What is the field on the axis of an infinite solenoid (infinite in both directions)? a FIGURE 5.25 Ꮎ, P Ꮎ, Solenoids and Helmholtz coils: 1. Griffiths 5.11. This is a finite solenoid problem that we already did in lecture, so you can reproduce the integration, but note that the angles defined in Fig. 5.24 in the textbook are different from the ones that we used in lecture. In addition to deriving the result (B field on axis everywhere), plot the B field for a solenoid of radius R and length L = 10 R. How close does the field at the center come to the field of an infinite solenoid μon? [Hint: we know the magnetic field of a ring of radius R carrying a current I on axis is given by B = µoIR²/[2(R² + z²)³/2], so breaking the solenoid into a bunch of rings should do the trick. The plot of B vs. position in the solenoid should look something like what I show below, but of course you will do a much better job: -5R Mon I 5R 2 = R is absent. 2. Now imagine a very, very long solenoid, but with a gap: a piece of length L What is the magnetic field right at the center of the gap? We will soon see that gapped solenoids are relevant to some practical problems. [Hint: think of superposition, the result of problem #1 might come in handy].