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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 11.3 The Biot-Savart Law

Definition 11.3.1. Magnetic field for a moving electric charge.

The magnetic field created by a moving electric charge is given by
\begin{equation*} \vec{B} = \frac{\mu_o}{4\pi}\frac{q\vec{v} \times \hat{r}}{r^2} \end{equation*}
where \(\vec{v}\) is the velocity of the moving charge and \(\vec{r}\) is the distance from the moving charge to the position where you are determining the magnetic field.

Exercises Activities

The figure below shows a proton moving to the right with speed \(v = 10^3\) m/s. The three labeled points all lie 2 m away from the proton at the instant shown.
Figure 11.3.2. Several points around a moving proton.

1. Explanation.

Predict the direction of the magnetic field at each point. Explain your reasoning.
Answer.
  1. the cross product is zero, so the magnetic field is also zero
  2. out of the screen
  3. out of the screen

2. Calculation.

Determine the magnitude of the magnetic field at each of the marked points. As usual, express your answers both symbolically and numerically.
Answer.
  1. \(\displaystyle B = 0 \text{ T}\)
  2. \(B = \frac{\mu_o qv}{4\pi r^2} = 4 \times 10^{-24}\) T
  3. \(B = \frac{\mu_o qv}{4\pi r^2}\frac{\sqrt{2}}{2} = 2.8 \times 10^{-24}\) T

3. Sensemaking: Covariational Reasoning.

How does the strength of the magnetic field change as you get farther away from the charge? Does it matter which direction you get farther away? Explain your reasoning.
Answer.
The magnetic field always gets smaller as you get farther away, but because of the cross product it also gets smaller as a result of moving in the direction the charge is moving.

4. Representation: Vector Field Map.

Use your answers to the above questions to sketch a magnetic field vector map for the region around the charge. Your map should highlight all the major features of the magnetic field you have identified so far.
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