AP Physics C: Electricity and Magnetism FRQ Practice

💡AP Physics C: Electricity and Magnetism

Practice FRQ 1 of 201/20

4. A point charge $+q = 3.0\,\mu\text{C}$ is held fixed on the $z$ -axis at $z = +0.20\,\text{m}$ above a large, flat boundary surface that lies in the $xy$ -plane at $z=0$ , as shown in Figure 1. The region $z>0$ is air with permittivity $\varepsilon_0$ . The region $z<0$ is filled with a linear dielectric material with permittivity $\varepsilon = 4\varepsilon_0$ . Ignore edge effects and assume the boundary is infinite. A small test charge $+q_0 = 2.0\,\text{nC}$ with mass $m = 1.0\,\text{g}$ is placed at point $P$ on the $z$ -axis at $z = +0.60\,\text{m}$ . The gravitational field is $g = 9.8\,\text{m/s}^2$ downward.

Figure 1. Point charge above an infinite planar boundary between air (ε0) and a dielectric (ε = 4ε0), with a test charge at point P on the z-axis.

Let $F_E$ be the magnitude of the electric force on the test charge $+q_0$ at point $P$ due to the source charge and induced charges at the boundary. Let $F_g$ be the magnitude of the gravitational force on the test charge.

Indicate whether $F_E$ is greater than, less than, or equal to $F_g$ by writing one of the following.

$F_E > F_g$
$F_E < F_g$
$F_E = F_g$

Justify your answer by identifying the direction of the electric force at $P$ and comparing the expected magnitudes using relevant physics relationships.

Derive an expression for the magnitude $E_P$ of the electric field at point $P$ in the air region. Use the method of images for a point charge above a planar interface between two linear media with permittivities $\varepsilon_1$ (for $z>0$ ) and $\varepsilon_2$ (for $z<0$ ).

Begin your derivation by writing a fundamental physics principle or an equation from the reference information. Your final expression should be in terms of $q$ , $\varepsilon_1$ , $\varepsilon_2$ , the Coulomb constant $k = \frac{1}{4\pi\varepsilon_0}$ , and the distances from $P$ to the real charge and to the image charge. Clearly indicate any superposition you use.

(For this configuration, the image charge magnitude is $q' = \left(\frac{\varepsilon_1-\varepsilon_2}{\varepsilon_1+\varepsilon_2}\right)q$ located the same distance below the boundary as the real charge is above it.)

Figure 2. Same charge locations as Figure 1, but the lower half-space is a conductor with a conducting plane at z = 0.

Indicate whether $E_{\text{cond}}$ is greater than, less than, or equal to $E_P$ by writing one of the following. Later, the dielectric region is replaced by a conductor, as shown in Figure 2. The charge $+q = 3.0\,\mu\text{C}$ remains fixed at $z = +0.20\,\text{m}$ , and point $P$ remains at $z = +0.60\,\text{m}$ in air. Let $E_{\text{cond}}$ be the magnitude of the electric field at $P$ for the conductor case, and let $E_P$ be the magnitude from part B for the dielectric case with $\varepsilon_2 = 4\varepsilon_1$ .

$E_{\text{cond}} > E_P$
$E_{\text{cond}} < E_P$
$E_{\text{cond}} = E_P$

Briefly justify your answer by referencing your expression from part B and the appropriate image-charge factor for a conducting plane.

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