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Chapter 26: Problem 69

Derive Equation 26.21 for the solenoid field by considering the solenoid to be made of infinitesimal current loops. Use Equation 26.9 for the loop fields, and integrate over all loops.

Short Answer

Expert verified
The magnetic field inside an ideal solenoid is given by \( B = \mu_0 n I \) where \( \mu_0 \) is the permeability of free space, \( I \) is the current flowing, and \( n \) is the number of turns per unit length.

Step by step solution

01

Use Biot-Savart Law

The first thing to do is recall the Biot-Savart law for the magnetic field generated by an infinitesimal current element. This law, by using the equation 26.9, can be written as: \[ d\vec{B}= \frac{\mu_0Id\vec{l} \times \vec{r}}{4\pi r^3} \]. Here, \( \mu_0 \) is the permeability of free space, \( I \) is the current flowing through the infinitesimal current loop, \( d\vec{l} \) is a vector element of the loop, and \( \vec{r} \) is a vector from the current element to the point where you're trying to find the field.
02

Compute the Magnetic Field using Symmetry

Due to the symmetry of the problem, the magnetic field due to an individual loop can be broken down into radial and axial components, with only the axial components summing up to give a net magnetic field. When integrated over the loop, this gives: \[ dB = \frac{\mu_0 I dl \sin\theta}{4\pi r^2} \] where \( \theta \) is the angle between \( d\vec{l} \) and \( \vec{r} \) and \( r \) is the distance from \( dl \) to the point where the magnetic field is calculated.
03

Integrate Over All Loops

To find the total field in the solenoid, the field due to each infinitesimal current loop needs to be added up. This is done by integrating over the length of the solenoid, as all the loops are identical: \[ B = \int dB = \int_{0}^{L} \frac{\mu_0 I dl \sin\theta}{4\pi r^2} = \frac{碌_0nI}{2} (cos洗1 - cos洗2) \] where \( n \) is the number of turns per unit length and \( 洗1 \) and \( 洗2 \) are the angles made by the start and end of the solenoid with the point where the magnetic field is being calculated.
04

Calculate for the Ideal Case

Considering an ideal solenoid, where \( 洗1 = 0 \) and \( 洗2 = 蟺 \), we replace these values in the equation obtained in Step 3. This derives the equation for magnetic field inside an ideal solenoid: \[ B = \mu_0 n I \]. This is the equation 26.21

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