AP Physics C: Electricity and Magnetism Unit 12 Review: Magnetic Fields & Electromagnetism

Derive an expression for the magnitude of the magnetic force F on the particle at point P and calculate its numerical value. Express your answer in terms of q, v, I, rP, μ0, and μr, as appropriate, and then evaluate using q = +2.0 μC, v = 3.0×10^5 m/s, I = 8.0 A, rP = 4.0 cm, μr = 250, and μ0 = 4π×10^-7 T·m/A. The charged particle passes through point P with velocity v = 3.0×10^5 m/s in the +y-direction, as shown in Figure 3. The magnetic field at P is the field produced by the current in the wire and the magnetic material configuration.

Describe a procedure for collecting data in Experiment 1 (air core) and Experiment 2 (unknown core) that would allow the students to use a graph to determine the relative permeability $\mu_r$ of the core material, including any steps necessary to reduce experimental uncertainty.

Describe how the collected data could be graphed and how that graph would be analyzed to determine $\mu_r$. In your answer, relate the graph to Ampère’s law for a long solenoid and identify how the slope (or other feature) is connected to $\mu_r$.

Using the best-fit line that you drew in part C(iii), calculate an experimental value for $\mu_r$. Assume the magnetic field inside a long solenoid is well approximated by $B = \mu\,n\,I$, where $n = \dfrac{N}{L}$ and $\mu = \mu_0\mu_r$. Use the solenoid parameters $N = 600$ turns and $L = 0.300\ \text{m}$, and use $\mu_0 = 4\pi\times10^{-7}\ \text{T\,m/A}$.

In Figure 2, draw bars to represent $|B|$ for (1) the core absent (air) and (3) the current reversed with the core present, relative to the reference bar shown for (2). If $|B| = 0$, write a "0" in that column. The magnitude of the magnetic field at the center of the solenoid is $|B|$. The partially completed bar chart in Figure 2 shows a bar that represents $|B|$ when the core is present with $\mu_r = 200$ and the current is $I = 2.0\ \text{A}$.

Derive an expression for the magnitude of the magnetic field $|B|$ at the center of the solenoid when the core is present, in terms of $N$, $L$, $I$, $\mu_0$, and $\mu_r$. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. Your derivation must include an explicit evaluation of $\oint \vec{B}·d\vec{\ell}$ for an appropriate Amperian loop and must connect the result to the definition of permeability for a linear material.

A proton of charge $+e$ enters the solenoid along the +y-direction at a position near the center where the field is uniform. The solenoid current is such that $\vec{B}$ inside points in the +x-direction. On the axes shown in Figure 3, sketch a graph of the x-component of the magnetic force $F_x$ on the proton as a function of the proton's position x as it travels from x = -0.10 m to x = +0.10 m along the axis. Assume the proton's speed remains constant and the only significant force is the magnetic force.

Indicate whether the direction of the magnetic force on the wire segment is +y or -y, and briefly justify your answer by referencing the vector relationship between $\vec{F}$, $\vec{I_w}$ (or $d\vec{\ell}$), and $\vec{B}$. Then calculate the magnitude of the force on the wire segment. A straight wire segment of length $\ell = 4.0\ \text{cm}$ is placed at the center of the solenoid, oriented along the +z-direction, and carries a current $I_w = 3.0\ \text{A}$ in the +z-direction. The solenoid has the core present with $\mu_r = 200$ and carries $I = 2.0\ \text{A}$. The magnetic field inside the solenoid is uniform and directed in the +x-direction.