Exam 29: Potential and Field

In a certain region, the electric potential due to a charge distribution is given by the equation V(x,y,z) = 3x^2y2 + yz³ - 2z^3x, where x, y, and z are measured in meters and V is in volts. Calculate the magnitude of the electric field vector at the position (x,y,z) = (1.0, 1.0, 1.0).

Each plate of a parallel-plate air-filled capacitor has an area of 0.0020 m², and the separation of the plates is 0.020 mm. An electric field of 3.9 x 10⁶ V/m is present between the plates. What is the surface charge density on the plates? (ε₀ = 8.85 × 10^-12 C²/N ∙ m²)

The charge on the square plates of a parallel-plate capacitor is Q. The potential across the plates is maintained with constant voltage by a battery as they are pulled apart to twice their original separation, which is small compared to the dimensions of the plates. The amount of charge on the plates is now equal to

Two capacitors, C₁ and C₂, are connected in series across a source of potential difference. With the potential source still connected, a dielectric is now inserted between the plates of capacitor C₁. What happens to the charge on capacitor C₂?

A conducting sphere of radius R carries an excess positive charge and is very far from any other charges. Which one of the following graphs best illustrates the potential (relative to infinity) produced by this sphere as a function of the distance r from the center of the sphere?

The graph in the figure shows the variation of the electric potential V (measured in volts) as a function of the radial direction r (measured in meters). For which range or value of r is the magnitude of the electric field the largest?

When two or more capacitors are connected in parallel across a potential difference

A parallel-plate capacitor has a capacitance of 10 mF and is charged with a 20-V power supply. The power supply is then removed and a dielectric material of dielectric constant 4.0 is used to fill the space between the plates. What is the voltage now across the capacitor?

An air-filled parallel-plate capacitor is connected to a battery and allowed to charge up. Now a slab of dielectric material is placed between the plates of the capacitor while the capacitor is still connected to the battery. After this is done, we find that

The electric field between square the plates of a parallel-plate capacitor has magnitude E. The potential across the plates is maintained with constant voltage by a battery as they are pulled apart to twice their original separation, which is small compared to the dimensions of the plates. The magnitude of the electric field between the plates is now equal to

An ideal air-filled parallel-plate capacitor has round plates and carries a fixed amount of equal but opposite charge on its plates. All the geometric parameters of the capacitor (plate diameter and plate separation) are now DOUBLED. If the original energy density between the plates was u₀, what is the new energy density?

The four identical capacitors in the circuit shown in the figure are initially uncharged. Let the charges on the capacitors be Q₁, Q₂, Q₃, and Q₄ and the potential differences across them be V₁, V₂, V₃, and V₄. The switch is thrown first to position A and kept there for a long time. It is then thrown to position B. Which of the following conditions is true with the switch in position B?

A conducting sphere contains positive charge distributed uniformly over its surface. Which statements about the potential due to this sphere are true? All potentials are measured relative to infinity. (There may be more than one correct choice.)

The capacitive network shown in the figure is assembled with initially uncharged capacitors. A potential difference, V_ab = +100V, is applied across the network. The switch S in the network is initially open but is then closed. Assume that all the capacitances shown are accurate to two significant figures. What is the equivalent capacitance between ab (a) with the switch S open? (b) with the switch S closed?

Three capacitors, of capacitance 5.00 μF, 10.0 μF, and 50.0 μF, are connected in series across a 12.0-V voltage source. (a) How much charge is stored in the 5.00-μF capacitor? (b) What is the potential difference across the 10.0-μF capacitor?

A 15-μF air-filled capacitor is connected to a 50-V voltage source and becomes fully charged. The voltage source is then removed and a slab of dielectric that completely fills the space between the plates is inserted. The dielectric has a dielectric constant of 5.0. (a) What is the capacitance of the capacitor after the slab has been inserted? (b) What is the potential difference across the plates of the capacitor after the slab has been inserted?

An ideal air-filled parallel-plate capacitor has round plates and carries a fixed amount of equal but opposite charge on its plates. All the geometric parameters of the capacitor (plate diameter and plate separation) are now DOUBLED. If the original energy stored in the capacitor was U₀, how much energy does it now store?

A metallic sphere of radius 5 cm is charged such that the potential of its surface is 100 V (relative to infinity). Which of the following plots correctly shows the potential as a function of distance from the center of the sphere?

The capacitive network shown in the figure is assembled with initially uncharged capacitors. A potential difference, V_ab = +100V, is applied across the network. The switch S in the network is kept open. Assume that all the capacitances shown are accurate to two significant figures. What is potential difference V_cd across the open switch S?

The capacitors in the network shown in the figure all have a capacitance of 5.0 µF. What is the equivalent capacitance, C_ab, of this capacitor network?