6.5: Electric Motors, Displacement Furrent, and EM Waves

Learning Objectives

• Determine the torque from an external magnetic field on a current loop.
• Identify magnetic dipole moment and how it relates to torque and potential energy.
• Model how electric motors convert electrical energy to rotational energy.

Model

Current runs through a square loop that is located between magnets that produce a constant magnetic field directed to the right.

Explore

1. Identify the direction (A to B or B to A) that current flows in each panel of this rotation.
2. Draw the magnetic moment \(\vec{\mu}\) on each panel using the Right Hand Rule.
3. Identify the coordinate direction (\(+/-\hat{x}\), \(+/-\hat{y}\), or \(+/-\hat{z}\)) of:

   	1	2	3
   \(\vec{\mu}\)
   \(\vec{B}\)
   \(\vec{\omega}\)

Critical Thinking & Concept Invention

1. Recall that torque is the cross product of the magnetic moment and the magnetic field:

   \[\vec{\tau}=\vec{\mu}\times\vec{B}\]

   Draw the torque vector on each panel. If the torque is zero, write:

   \[\vec{\tau}=0\]

2. Identify the coordinate direction (\(+/-\hat{x}\), \(+/-\hat{y}\), or \(+/-\hat{z}\)) of:

   	1	2	3
   \(\vec{\tau}\)
   \(\dfrac{d\vec{\omega}}{dt}\)

3. Identify whether the loop’s rotation is speeding up, slowing down, or neither at each moment. Explain your reasoning.
4. Is this an effective motor? Explain why or why not.

Model

Explore

1. Identify the direction (A to B or B to A) that current flows in each panel. If there is no current, write “No Current.”
2. Draw the magnetic moment \(\vec{\mu}\) on each panel using the Right Hand Rule.
3. Explain what is different about this scenario from the previous one.

Critical Thinking & Concept Invention

1. Recall:

   \[\vec{\tau}=\vec{\mu}\times\vec{B}\]

   Draw the torque vector on each panel. If the torque is zero, write:

   \[\vec{\tau}=0\]

2. Identify the coordinate direction (\(+/-\hat{x}\), \(+/-\hat{y}\), or \(+/-\hat{z}\)) of:

   	1	2	3
   \(\vec{\tau}\)
   \(\dfrac{d\vec{\omega}}{dt}\)

3. Identify whether the loop’s rotation is speeding up, slowing down, or neither at each moment. Explain your reasoning.
4. Is this an effective motor? Explain why or why not.
5. Explain what must happen in a motor to continuously increase the rotation rate.

Summary

In a DC motor, the current switches direction through the circuit every half-rotation. This flips the magnetic moment vector and ensures that the torque remains in the same direction as the angular velocity, continuously increasing the rotational speed.

Apply

A square loop with current \(I\) and side length \(L\) rotates in a constant magnetic field of strength \(B\). At a particular instant, the loop has an area vector in the \(x\)-\(y\) plane at an angle \(\theta\) with respect to the \(x\)-axis.

• Identify the torque vector at this instant.
• Plot the torque on the loop as a function of \(\theta\) from \(\theta=0\) to \(\theta=\pi\), both with and without a commutator.

Model

The model shows a capacitor with current flowing into and out of the plates. Two Ampere surfaces share the same closed boundary \(C\). The model on the right is incomplete.

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

Explore

1. Describe each shape.
2. Identify the enclosed current through each surface.
   • \(I_1=\)
   • \(I_2=\)
3. Identify the electric flux through each surface.
   • \(\Phi_{E1}=\)
   • \(\Phi_{E2}=\)
4. If current flows as shown, will there be more or fewer positive charges on the top plate over time? Draw charges accordingly.
5. Based on your answer, describe how the electric field magnitude and electric flux change:
   • The electric field and electric flux through \(S_1\) will...
   • The electric field and electric flux through \(S_2\) will...

Critical Thinking & Concept Invention

1. Ampère’s Law states that magnetic circulation depends on enclosed current. Which surface suggests that the magnetic field should be zero along curve \(C\)?
2. Using the other surface, determine the magnetic field along curve \(C\):

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

3. For the surface with zero enclosed current, which quantity was changing: \(\Gamma\) or \(\Phi\)? Include the appropriate electric or magnetic subscript.
4. Maxwell proposed a new magnetic circulation equation.
   • Identify the displacement current \(I_d\).
   • Write the complete magnetic circulation equation.
   • Verify that it works for both surfaces by identifying \(I_{\mathrm{enc}}\) and \(I_d\) for each.

   \[\Gamma_B=\mu_0\left(I_{\mathrm{enc}}+I_d\right)\]

   \[I_d=\]

   \[\Gamma_B=\]

Summary

The equation developed above is known as the Maxwell–Ampère Law, and the displacement current is Maxwell’s correction to Ampère’s Law.

\[I_d\equiv\]

\[\Gamma_B=\mu_0\left(I_{\mathrm{enc}}+I_d\right)\]

\[\Gamma_B=\]

Displacement current is not an actual flow of charge. Instead, it is a changing electric flux that produces the same magnetic field that an ordinary current would produce. The electric constant \(\epsilon_0\) is included and often cancels a corresponding factor appearing in electric flux.

Apply

The potential difference \(V(t)\) between parallel plates increases at a rate of \(10^7\ \mathrm{V/s}\). The plates are separated by \(1.00\ \mathrm{cm}\) and have an area of \(0.200\ \mathrm{m}^2\).

• Determine the electric field as a function of time, assuming very large plates.
• Determine the electric flux as a function of time.
• Determine the displacement current.

Model

A current varies periodically with time, such as: \(I(t)=I_0\sin(\omega t)\). At the instant shown, the current is directed upward and increasing.

Explore

1. Identify the direction of the magnetic field created by the current by filling in the into-page or out-of-page symbols on the Ampere loop.
2. A circle labeled \(\vec{E}\) intersects one magnetic field point. Determine:
   • Does the magnetic flux point into or out of the page?
   • Is the magnetic flux increasing or decreasing?
3. Determine the direction of the electric field circulation around the circle and draw arrows.

Critical Thinking & Concept Invention

1. Verify that all annotations on the left diagram are consistent.
2. Copy the electric field circulation onto the leftmost electric field circle on the right diagram.
3. Determine the direction of electric flux through the Ampere loop intersected by the electric field.
4. Assume the curling electric field is increasing. Draw the displacement current and the magnetic field it creates.
5. Assume the new magnetic field is decreasing. Draw the electric field circulation produced by the changing magnetic flux.
6. Assume the new electric field is decreasing. Draw the magnetic field circulation produced by the changing electric flux.
7. Summarize each step and explain how a changing current creates changing magnetic and electric fields at different points in space.

Summary

Gauss’s Law and Ampère’s Law show that electric and magnetic fields originate primarily from charges and currents. Faraday’s Law and Maxwell’s correction to Ampère’s Law demonstrate that changing electric and magnetic fluxes can also generate fields. This allows changes in one field to create changes in the other indefinitely. The graphic below is for a field where the flux was always increasing:

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

Light is an electromagnetic wave. A periodic source changes one field, which changes its associated flux, producing the other field. This sequence propagates away from the source. The electric and magnetic fields remain offset by \(90^\circ\) because each is generated by changes in the other.

Solving Maxwell’s Wquations gives:

\[\frac{\partial^2E}{\partial x^2}=\epsilon_0\mu_0\frac{\partial^2E}{\partial t^2}\]

\[\frac{\partial^2B}{\partial z^2}=\epsilon_0\mu_0\frac{\partial^2B}{\partial t^2}\]

Equations of this form (a partial second spatial derivative = a partial second time derivative) are wave equations. The coefficient equals the inverse square of the propagation speed (wave speed):

\[\epsilon_0\mu_0=\frac{1}{v^2}\]

\[v=\frac{1}{\sqrt{\epsilon_0\mu_0}}\]

Because both \(\epsilon_0\) and \(\mu_0\) are constants in vacuum, the speed of light is also constant:

\[c=\frac{1}{\sqrt{\epsilon_0\mu_0}}=3\times10^8\ \mathrm{m/s}\]

In other materials, the speed changes because \(\epsilon\) and \(\mu\) differ from their vacuum values.