AP Unit 10 Problems

Find the charges on A and B, explain the sign of each, and justify the point-charge approximation. Include the charge of a neutron and why the charges are quantized.

Find the magnitude and direction of each mutual force. Then predict the force factor if both charge magnitudes double and the separation triples. Also explain what changes if only B’s charge sign reverses.

Derive FE/Fg for a proton pair, calculate it, and explain why the planetary claim does not follow. Explain how electric interactions also relate to the normal force of a table.

For fixed free charges in an ideal uniform linear dielectric with κ = 5, predict the electric-force factor relative to vacuum. Explain what permittivity measures, polarization, the role of composition, and why an insulator can respond without conducting across the sample.

Find the cloth charge and the number of electrons transferred during rubbing. Explain why rubbing changes net charge but bringing the rod nearby alone does not change the can’s net charge.

Draw and describe the signs on the can’s near and far regions, then predict how a neutral insulating strip could also be attracted.

Design a sequence that leaves the sphere with negative net charge without touching the rod to it. Illustrate the charge distributions and identify electron-transfer direction and the step at which net charge changes.

In a separate contact experiment, remove the rod and all other asymmetric external fields. Two identical metal spheres, initially −4 nC and +10 nC, touch in a symmetric environment while isolated from Earth, then separate. Find their final charges and each signed charge change.

Draw the separate source-field arrows at P and calculate the net field. Explain what arrow length represents in a vector map and what field-line density would imply. Then reverse only the right-hand source to −2.0 nC: draw its new source-field arrow and calculate the new net field at P.

Find the force on a small −3.0 nC test charge at P and compare it with the force on +3.0 nC at the same point in the original two-positive-source configuration. Explain why making either test charge nonperturbing matters.

Calculate the field at r = 0.030 m and r = 0.10 m from its center. Sketch E magnitude against r and state the direction at its surface. State where the sphere’s excess charge resides at equilibrium.

Explain why an internal field would contradict equilibrium in the metal. Replace the sphere with an insulating sphere carrying excess charge through its volume: can you conclude its internal field is zero?

Find Ui, Uf, and ΔU. State the external work and electric-force work if the speeds stay negligibly small.

Sketch the U versus r curve for this pair, label zero at infinity, and compare the energy change if both separations were doubled.

Derive the total assembly energy by listing the pairs. For q = 4.0 nC and a = 0.20 m, calculate it.

The negative charge is moved infinitely far away while the two positive charges stay fixed. Find ΔU and the slow external work symbolically and numerically.

Find V(P), then the interaction energy of a +3.0 nC test charge at P. Explain why source-field arrows alone cannot be added as potential arrows.

A 9 V battery moves 2.0 mC of positive charge internally from its low-potential terminal to its high-potential terminal. Find the energy increase and name its source. Separately, two unequal isolated metal spheres touch: what equality must hold after redistribution?

Draw local field arrows in both regions with relative magnitudes. Explain their orientation, direction, and whether a field component lies along a contour.

A −2.0 μC charge moves from the 24 V line to the 12 V line. Find ΔU and describe its initial electric-force direction. What is the work if it instead moves between two points on the 18 V line?

Find C, each plate’s charge, central field magnitude and direction, and stored energy. Explain why Q is not the capacitor’s net charge.

Construct a quantitative graph of capacitor ΔV (V) against Q (nC), using numerical scales and plotting (0, 0), (1.77 nC, 5 V), and (3.54 nC, 10 V) as it charges from zero. Explain why its area is the stored energy ½QΔV and distinguish that from the work supplied by the constant 10 V battery. Then predict C and U if overlap area doubles with the battery still connected.

Compare final charge, voltage, field factor, and stored energy for dielectric insertion with the battery disconnected and with it connected. Explain the induced field direction.

A particle with q = −1.0 μC and m = 2.0 × 10⁻⁴ kg enters horizontally at 4.0 m/s into the downward 500 N/C field over horizontal length 0.20 m. Create a qualitative y-versus-x graph of the path with labeled axes (x rightward, y upward) and indicate its horizontal initial tangent. Then find the electric acceleration, transit time, and deflection. Neglect gravity and edge fields, and assume no plate collision.

Find ΔU, final kinetic energy in joules and electron-volts, and final speed. Draw energy bars that represent the change.

A slow external agent instead transports the same proton from 100 V to 300 V at unchanged speed. Find Wext and explain why final kinetic energy no longer follows −qΔV alone.

Derive the potential at the one-dimensional turning point and explain the meaning of a calculated negative final kinetic energy at a proposed higher potential.

Two particles have equal mass and equal initial speed, but charges +q and +2q. Compare their maximum potential increases. Then describe a negative particle’s energy change moving to higher potential.

Construct V versus x, estimate the average field over the whole interval, and give a conservative relative uncertainty using δE/E ≈ δ(ΔV)/|ΔV| + δ(Δx)/Δx. State a control and an assumption of this map.

Draw equipotentials and field arrows consistent with these data if the region is uniform vertically. Predict force on q = −2.0 μC and ΔU if it moves from x = 0 to x = 0.06 m under a corresponding static source field.

Choose a graph that linearizes the geometry relation, construct it with numerical scales and units, plot the transformed data, fit a line, determine its slope, and infer κ. Outline how to measure C versus d with a capacitance meter, including one control, one repeat strategy, and one likely systematic error.

At d = 2.0 mm, charge the capacitor to 10 V, disconnect it, and then double d. Find final Q, V, E, and U. Explain the charge conservation and external energy input in a slow separation.

Find the source field and potential at the origin, and the interaction energy associated with adding q there. Explain why the field and potential results do not conflict.

Give q a small initial velocity along the perpendicular y axis. If it eventually reaches very far away, derive its final speed in terms of m, vi, Q, q, and a. What can you say about its kinetic-energy change before knowing its travel time?