AP Physics C: Electricity and Magnetism Unit 9 Review: Electric Potential

$V_A$ is the electric potential at point A due to Charges 1 and 2. $V_B$ is the electric potential at point B due to Charges 1 and 2.

Derive an expression for the x-component $E_x(B)$ of the electric field at point B due to Charges 1 and 2 in terms of $k$, the charge magnitude $q = 3.0\ \text{nC}$, and the given distances. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Indicate whether $\left|E_{x,\text{new}}(B)\right|$ is greater than, less than, or equal to $\left|E_x(B)\right|$ by writing one of the following. Later, Charge 2 is changed to $-6.0\ \text{nC}$, but it remains at $x=+0.20\ \text{m}$, as shown in Figure 2. Charge 1 remains $+3.0\ \text{nC}$ at $x=-0.20\ \text{m}$. Let $E_{x,\text{new}}(B)$ be the new x-component of the electric field at point B.

Derive an expression for the speed $v_B$ of the particle at point B in terms of $q_0$, $m$, $q_1$, $q_2$, $x_A$, $x_B$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. A particle with charge $q_0 = +2.0\,\mu\text{C}$ and mass $m = 1.5× 10^{-4}\,\text{kg}$ moves along the x-axis from point A at $x_A=0.00\,\text{m}$ to point B at $x_B=+0.60\,\text{m}$, as shown in Figure 3. The particle is released from rest at point A and only electric forces act on the particle during the motion.

In Figure 2, draw bars to represent the electric potential $V$ at points A, B, and C relative to $V_P$ shown at point P. The sign of the potential must be indicated by drawing the bar above (positive) or below (negative) the horizontal axis. If $V = 0$ at a point, write a "0" in that column. The electric potential at point P is $V_P$. The partially completed bar chart in Figure 2 shows a bar that represents $V_P$ as a positive value.

Derive an expression for the x-component of the electric field, $E_x(x)$, at a general point on the x-axis in the region $-0.30\,\text{m} < x < +0.30\,\text{m}$ in terms of $q_1$, $q_2$, $x$, and physical constants, as appropriate. Begin your derivation by writing an expression for the electric potential $V(x)$ due to the two charges and then relate $E_x$ to the spatial rate of change of $V$.

On the axes shown in Figure 3, sketch a graph of the electric potential $V(x)$ as a function of position x for $-0.80\,\text{m} ≤ x ≤ +0.80\,\text{m}$ due to the two charges. The sketch must show the correct qualitative behavior near $x=-0.30\,\text{m}$ and $x=+0.30\,\text{m}$ and must be consistent with the relative values of $V$ at points A, P, B, and C from part A.

Indicate whether the electric potential energy of the test charge increases, decreases, or remains the same as it moves from P to C. Then calculate the change in electric potential energy $\Delta U$ of the test charge for the move from P to C. Briefly justify your answer by referencing the relationship between electric potential difference and potential energy. A test charge of magnitude $q_t = +1.50\,\text{nC}$ moves along the x-axis from point P at $x=0.00\,\text{m}$ to point C at $x=+0.60\,\text{m}$. Use $k = 8.99\times10^9\,\text{N}\,\text{m}^2/\text{C}^2$.