AP Physics 2 Unit 12 Review: Magnetism and Electromagnetism

A student claims that the magnitude of the magnetic force on the proton at point $P$ is greater when the core material has $\mu_r = 200$ than when the solenoid contains only air (approximately $\mu_r = 1$).

Derive an expression for the magnitude of the magnetic force on the proton at point $P$. Express your answer in terms of $N$, $L$, $I$, $\mu_r$, $v$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Indicate whether the expression you derived in part B is or is not consistent with your answer from part A. Briefly justify your answer by describing how the force depends on $\mu_r$.

On the dot shown (Figure dot), representing the particle, draw and label the magnetic force that is exerted on the particle. The force must be represented by a distinct arrow starting on, and pointing away from, the dot. A uniform magnetic field of magnitude $0.30\ \text{T}$ points to the right. A positively charged particle with charge $+2.0\ \mu\text{C}$ moves upward with speed $3.0× 10^5\ \text{m/s}$ through the field.

Derive an expression for the magnitude of the magnetic field $B$ inside the solenoid when it carries a steady current $I$, in terms of $N$, the solenoid length $L$, $I$, the permeability of free space $\mu_0$, and the relative permeability $\mu_r$ of the material filling the solenoid. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

On the axes provided (Figure 2), sketch the expected relationship between the magnetic field magnitude $B$ inside the solenoid and time $t$ for the interval $0.0\ \text{ms} ≤ t ≤ 6.0\ \text{ms}$. Clearly indicate any intervals of constant $B$ and any intervals during which $B$ changes. Do NOT calculate numerical values. The solenoid is initially empty (air-filled, $\mu_r = 1.0$) and the switch is open. At $t = 0.0\ \text{ms}$ the switch is closed, and the current in the solenoid increases linearly from $0.0\ \text{A}$ to $2.0\ \text{A}$ during the interval $0.0\ \text{ms} ≤ t ≤ 2.0\ \text{ms}$. The current then remains constant at $2.0\ \text{A}$ for $2.0\ \text{ms} < t < 4.0\ \text{ms}$. At $t = 4.0\ \text{ms}$, a ferromagnetic core with constant relative permeability $\mu_r = 500$ is inserted fully into the solenoid, and the insertion takes $2.0\ \text{ms}$, ending at $t = 6.0\ \text{ms}$. During the core insertion, the current remains constant at $2.0\ \text{A}$.

Indicate whether the magnitude of the induced potential difference across the loop is greater during Interval 1, greater during Interval 2, or the same during both intervals. The loop has one turn and radius $0.040\ \text{m}$. Treat the magnetic field as uniform over the area of the loop and equal to the field inside the solenoid. Consider two time intervals:

Indicate whether there is a clockwise induced current in the loop, a counterclockwise induced current in the loop, or no induced current in the loop. A single circular conducting loop of radius $R = 1.0\ \text{cm}$ is placed in the plane of the page with its center a distance $r_c = 4.0\ \text{cm}$ above the long straight wire, as shown in Figure 4. The loop is then translated upward (in the +y-direction) at a constant speed $v_{\text{loop}} = 0.50\ \text{m/s}$ while the current in the wire remains constant. Assume the magnetic field produced by the wire is approximately uniform over the area of the loop and is equal to the wire's magnetic field evaluated at the loop's center (i.e., at distance $r = r_c$ from the wire).