6.2: Solenoids and Toroids

Learning Objectives

• Use Ampère’s Law to determine magnetic fields from currents using integrals when needed.
• Use Ampère’s Law to calculate the magnetic field in regions of space around a solenoid.
• Use Ampère’s Law to calculate the magnetic field in regions of space around a toroid.

Model

Explore

1. Identify the general direction you expect the magnetic field to be along each path of the loop using the Right Hand Rule. Draw arrows to show this direction.
2. What is the enclosed current in the dashed Ampère loop?
3. Fill out the table to compare the path and magnetic field.

Critical Thinking & Concept Invention

1. Use Ampère’s Law to explore the magnetic fields along this loop.
   • Write a circulation equation for the loop as a sum of integrals for each segment, including any that are zero.
   • Take the dot products and move any constants outside the integral.
   • Evaluate each integral.
   • Separately solve for each nonzero magnetic field using appropriate subscripts.
   • In each equation, treat the other magnetic field and radius as constants and examine how magnetic field depends on radius.

   \[ \Gamma_B=\mu_0 I_{\mathrm{enc}} \]

   \[ B= \]

   \[ B= \]

   \[ B\propto r \qquad\text{OR}\qquad B\propto\frac{1}{r} \]

2. Why was it useful to choose an Ampère loop that did not enclose current?

Model

Explore

1. Identify the general direction you expect the magnetic field to be along each loop using the Right Hand Rule.
2. Identify the enclosed current in each Ampère loop.

   \[ I_1=\qquad I_2= \]

3. What should the magnetic circulation be in each loop?

   \[ \Gamma_1=\qquad \Gamma_2= \]

Critical Thinking & Concept Invention

1. Use Ampère’s Law to explore the magnetic field along Loop 1.
   • Write the circulation equation as a sum of integrals for each segment.
   • Take the dot products and move constants outside the integral.
   • Evaluate each integral.
   • Solve for one of the nonzero magnetic fields using appropriate subscripts.

   \[ \Gamma_B=\mu_0 I_{\mathrm{enc}} \]

   \[ B= \]

2. What must be true about the magnetic field at different places within the solenoid to make this equation true? Explain.
3. Use Ampère’s Law to explore the magnetic field along Loop 2.
   • Write the circulation equation as a sum of integrals for each segment.
   • Take the dot products and move constants outside the integral.
   • Evaluate each integral.
   • Solve for one of the nonzero magnetic fields using appropriate subscripts.

   \[ \Gamma_B=\mu_0 I_{\mathrm{enc}} \]

   \[ B= \]

4. If the farthest side of Loop 2 is placed very far away, what should the magnetic field be along that far edge?
5. Based on your previous response, what is the magnitude of the magnetic field on the near edge, which can be anywhere outside the solenoid? Explain your reasoning.

Summary

In a solenoid:

• The magnetic field inside is __________________.
• The magnetic field outside is __________________.

Model

Explore

1. Identify the general direction you expect the magnetic field to be along each path of the loop using the Right Hand Rule. Draw arrows to show this direction.
2. What is the enclosed current in the dashed Ampère loop? How would this change if the loop height were greater or smaller?
3. Label each segment as \(\Delta l_1\), \(\Delta l_2\), and so on.
4. Fill out the table comparing the path and magnetic field.

Critical Thinking & Concept Invention

1. Use Ampère’s Law to explore the magnetic field along Loop 1.
   • Write the circulation equation as a sum of integrals for each segment.
   • Take the dot products and move constants outside the integral.
   • Evaluate each integral.
   • Solve for one of the nonzero magnetic fields.

   \[ \Gamma_B=\mu_0 I_{\mathrm{enc}} \]

   \[ B= \]

2. Generalize your equation.
   • For a different-sized loop, replace \(I_{\mathrm{enc}}\) with a variable using \(N\) for the number of coils.
   • Coil density is defined as \(n=N/L\). Construct a new equation for the magnetic field of a solenoid in terms of \(n\).

   \[ B= \]

   \[ B= \]

Model

A toroid is a solenoid bent into a donut shape. The toroid shown carries current \(I\), has \(N\) loops, an inner radius \(a\), and an outer radius \(b\).

Image Credit: OpenStax University Physics Vol 2 (https://openstax.org/details/books/university-physics-volume-2)

Explore

Examine the middle image:

1. Compare the magnetic field magnitudes at points \(P_1\), \(P_2\), and \(P_3\). Circle the equality or rank them using greater-than symbols.
2. Draw magnetic field vectors at each point based on their relative magnitudes. Use the Right Hand Rule to determine direction.
3. How does the direction of the magnetic field compare at each point?

Examine the image on the right:

\[ \Delta l_1=\Delta l_2=\Delta l_3 \qquad\text{OR}\qquad > > \]

4. How much current is enclosed in each Ampère loop?

   \[ I_1=\qquad I_2=\qquad I_3= \]

5. Draw a radius vector to each Ampère loop and label those vectors appropriately.
6. Compare the total lengths of the three Ampère paths. Circle the equality or rank them.
7. Compare the direction of the magnetic field to the path direction for each loop.

   \[ \vec{B}\perp d\vec{l} \qquad\text{OR}\qquad \vec{B}\parallel d\vec{l} \qquad\text{OR}\qquad \text{neither} \]

Critical Thinking & Concept Invention

1. Explore the magnetic fields along paths \(D_1\) and \(D_3\).
   • Write an Ampère’s Law equation for each path.
   • Simplify the dot product.
   • Move constants outside the integral.
   • Solve for \(B_1\) and \(B_3\).

   \[ \Gamma_1=\mu_0 I_1 \qquad \Gamma_3=\mu_0 I_3 \]

   \[ B_1= \qquad B_3= \]

2. What can you conclude about the magnetic field outside the toroid? Explain how you know this is true.
3. Explore the magnetic field along path \(D_2\).
   • Write an Ampère’s Law equation for this path.
   • Simplify the dot product.
   • Move constants outside the integral.
   • Solve for \(B_2\).

   \[ \Gamma_2=\mu_0 I_2 \]

   \[ B_2= \]

4. Why is the result different for \(D_2\) than for \(D_1\) and \(D_3\)?
5. Is the magnetic field constant throughout a toroid as it is in a solenoid? If not, where is the field strongest? Explain how the equation demonstrates this.