Draw a simplified top-view schematic based on the central portion of the apparatus shown. Your drawing must include the suspension axis, the horizontal arm, spheres $A$ and $B$, the distance $L$, the separation $r$, the electric force on sphere $A$, and the direction of the restoring torque.

Derive an expression for the magnitude of the charge $q$ in terms of $\kappa$, $\theta$, $r$, $L$, and $k$. Begin with the rotational-equilibrium condition and Coulomb’s law.

Calculate the charge on each sphere for the trial described.

Determine the new equilibrium angle if the separation is increased from $r$ to $2r$ while all other quantities remain unchanged.

Determine whether the electric potential energy of the two-sphere system increases, decreases, or remains constant when sphere $B$ is moved quasistatically closer to sphere $A$.

Justify your answer using an appropriate physical relationship or a work-energy argument.

Draw energy-bar representations for the sphere-particle system when the particle is at point $A$ and when it reaches point $B$. Include electric potential energy, kinetic energy, and total mechanical energy.

Derive an expression for the speed $v_B$ of the particle when it reaches point $B$ in terms of $k$, $Q$, $q$, $m$, and $R$.

Sketch a graph of electric potential $V$ as a function of radial distance $r$ for $r\geq R$. Label the values of the potential at $r=R$, $r=2R$, and $r=4R$, and show the limiting behavior at large $r$.

Determine the ratio $\dfrac{E_A}{E_B}$ of the electric-field magnitude at point $A$ to that at point $B$.

Justify why the relative spacing of the equipotential lines in the diagram is consistent with both the graph in part (c) and the ratio in part (d).

A particle with charge $-q$ is instead released from rest at point $B$. Determine the direction of its initial acceleration.

Indicate the independent variable and the dependent variable for an investigation of how plate separation affects capacitance.

Describe a repeatable experimental procedure. Include the equipment needed, how the relevant quantities would be measured, and at least two variables that should be controlled.

Describe a graphical analysis that would test the expected functional relationship between capacitance and plate separation and allow the students to determine the permittivity of free space.

Calculate the value of $\dfrac{1}{d}$ for Trial 3 in units of $\mathrm{m^{-1}}$.

Plot capacitance $C$ on the vertical axis as a function of $\dfrac{1}{d}$ on the horizontal axis. Label both axes with units, use sensible linear scales, plot all six data points, and draw an appropriate best-fit line.

Determine the experimental value of the permittivity of free space from the slope of the best-fit line.

Determine the direction of the electric field between the plates that allows the negatively charged drop to remain stationary.

Justify your answer by relating the directions of the electric force, the electric field, and the gravitational force on the drop.

Derive an expression for the magnitude of the potential difference required to suspend the drop in terms of $m$, $g$, $d$, and $q$.

The suspended drop suddenly gains additional electrons so that its charge becomes $-2q$, while its mass and the potential difference remain unchanged. Determine the magnitude and direction of the drop’s initial acceleration in terms of $g$.

Determine the slope of the best-fit line.

Indicate the SI units of the slope.

Let $s$ represent the slope of the graph. Derive an expression for $q_2$ in terms of $s$, $k$, and $q_1$.

Calculate the magnitude of $q_2$.

Draw a free-body diagram for the second sphere when it is to the right of the fixed sphere and no other forces act horizontally.

Calculate the acceleration of the second sphere when $r=0.25,\mathrm{m}$.

Justify whether the plotted results are consistent with the distance dependence predicted by Coulomb’s law.

Determine the radius of the sphere and the electric potential at its surface.

Draw a cross-sectional representation of the conducting sphere that includes the distribution of excess charge and electric-field vectors inside and outside the sphere.

Let the surface potential be $V_s$ and the sphere radius be $R$. Derive the relationship $E_s=V_s/R$ for the electric-field magnitude immediately outside the surface.

Calculate the electric-field magnitude immediately outside the sphere.

Sketch a graph of electric-field magnitude $E$ as a function of $r$ from $r=0$ to $r=0.50,\mathrm{m}$.

Justify how the electric-potential graph and the electric-field representation are consistent with each other both inside and outside the conductor.

Determine the capacitance of the capacitor represented by the graph.

Describe an experimental procedure to investigate how the capacitance of a parallel-plate capacitor depends on the overlapping plate area. Include the equipment, the independent and dependent variables, important controls, repeated measurements, and the graphical analysis.

Calculate $1/d$ for Trial $4$ in $\mathrm{m^{-1}}$.

Plot $C$ as a function of $1/d$ using all six observations, and draw a best-fit line.

Determine the slope of the best-fit line.

Calculate the overlapping plate area.

Determine whether the particle moves toward increasing or decreasing electric potential as its kinetic energy increases.

Justify the relationship between the direction of motion and the change in kinetic energy using electric potential energy and conservation of energy.

Let the initial potential be $V_i$ and the particle’s charge be $q$. Derive an expression for the kinetic energy $K$ when the particle is at potential $V$.

Determine the particle’s charge, including its sign.

Determine the direction in which a particle of charge $-q$ would initially accelerate if released from rest at the same starting location, and justify the direction.

Determine the slope of each plotted line.

Let $s$ represent the slope of an $E$-versus-$Q$ graph. Derive an expression for $s$ in terms of $\kappa$, $\epsilon_0$, and $A$.

Calculate the plate area using the line for Medium A.

Calculate the dielectric constant $\kappa_B$.

Draw the plates, the electric-field direction, and the direction of the electrostatic force on the particle.

Calculate the particle’s acceleration.

Justify why the electric field is smaller with Medium B than with Medium A for the same free plate charge.

Determine the magnitude and direction of the electric field in the air gap.

Draw a side-view schematic showing the two conducting plates, the signs of their surface charges, at least three equipotential lines in the gap, and electric-field vectors.

Let the potential difference magnitude between the plates be $\Delta V$ and the gap width be $d$. Derive an expression for the magnitude of the particle’s acceleration.

Calculate the particle’s acceleration.

Sketch the particle’s horizontal velocity $v_x$ as a function of time from release until it reaches the right plate.

Compare the initial acceleration of this particle with that of a particle having charge $-q$ and the same mass, released from rest at the same location.

Justify how the shape of the potential graph is consistent with the electric field inside the conductors and in the gap.