AP Physics C: Electricity & Magnetism 12.4 A Use Amp Res Law to Describe the Magnetic Field Created By a Moving Charge Carrier Questions

Use Ampere's law, symmetry, enclosed current, and superposition to determine magnetic fields of straight wires, cylindrical conductors, coaxial cables, and solenoids.

Exam points

choose and draw an Amperian loop that follows the symmetry of the current distribution
evaluate the magnetic-field line integral and the signed current enclosed by a chosen loop
derive magnetic-field magnitude and direction inside or outside a long straight conductor
integrate current density to find enclosed current within solid or hollow cylindrical conductors
derive piecewise magnetic-field expressions and graphs across conductor boundaries
analyse coaxial or concentric conductors and identify regions where enclosed currents cancel
derive and apply the uniform interior magnetic field of an ideal long solenoid
superpose fields from several long wires and locate points where the net field is zero
use a magnetic-field graph or best-fit slope to infer current, resistance, or vacuum permeability
design and evaluate a solenoid or wire-field experiment, including systematic offsets and percent error

[Maximum number: 3]

Long, parallel wires S and T are a distance 2 d apart. Both wires carry equal currents I, but the currents are in opposite directions. Both wires are parallel to the x-axis. At the instant shown in Figure 1, Sphere 1 is a distance d above Wire S, Sphere 2 is a distance d below Wire S, and both spheres are moving with speed v in the +x-direction. Each sphere has positive charge +Q.

Gravitational effects are negligible.

Figure 1

Derive an expression for the magnitude $B_{\text {tot }}$ of the magnetic field at the location of Sphere 2 due to the currents in wires S and T in terms of d, I, and physical constants, as appropriate.

Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

\multirow[t]{4}{*}{B} &

For a multistep derivation that includes the equation $\oint \vec{B} \cdot d \vec{\ell}=\mu_{0} I_{\text {enc }}$ or $d \vec{B}=\frac{\mu_{0}}{4 \pi} \frac{I(d \vec{\ell} \times \hat{r})}{r^{2}}$
Scoring Note: Vector notation is not required for this point to be earned. A multistep derivation that includes $B=\frac{\mu_{0} I}{2 \pi d}$ can earn this point.

& Point B1 \\ \hline & For a correct expression for the magnitude of the magnetic field due to the current in one wire at the location of Sphere 2 & Point B2 \\ \hline & For indicating that the magnitude of the total field is twice that due to one wire & Point B3 \\ \hline &

Example Response
Determine the magnitude B of the magnetic field due to one wire. $\oint \vec{B} \cdot d \vec{\ell}=\mu_{0} I_{\mathrm{enc}}$ $\begin{aligned} & B(2 \pi d)=\mu_{0} I & B=\frac{\mu_{0} I}{2 \pi d} \end{aligned}$
Determine $B_{\text {tot }}$ . $\begin{aligned} & B_{\text {tot }}=\frac{\mu_{0} I}{2 \pi d}+\frac{\mu_{0} I}{2 \pi d} & B_{\text {tot }}=\frac{\mu_{0} I}{\pi d} \end{aligned}$

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AP Physics C: Electricity & Magnetism 12.4 A Use Amp Res Law to Describe the Magnetic Field Created By a Moving Charge Carrier Questions question 1