Capacitor and Circuit Problems: Solutions and Analysis
Problem 1: Spherical Capacitor
A spherical capacitor consists of a thin conducting shell of radius a, surrounded by a thin conducting shell of radius b (where b > a). When the capacitor is connected to a battery, the inner shell has charge +Q and the outer shell has charge –Q.
(a) Let r denote the distance from the center of the shells. Use Gauss’s law to find a symbolic expression for the electric field between the shells.
(b) Find a symbolic expression for the magnitude of the potential difference (voltage) between the two shells.
(c) Find a symbolic expression for the capacitance (recall that C = Q/V).
(d) Suppose the radii Ra = 12.5 cm and b = 17.5 cm. Calculate the capacitance, in units of Farads.
Problem 2: Cylindrical Capacitor
A cylindrical capacitor consists of a conducting cylinder of radius a, surrounded by an outer shell of radius b. Both cylinders have length L, which is much larger than the two radii (L >> a and L >> b).
(a) Let r denote the distance from the center of the shells. Use Gauss’s law to find a symbolic expression for the electric field between the shells.
(b) Find a symbolic expression for the electric energy density in between the two shells, in terms of r.
(c) Integrate your expression from part (b) over the volume of the capacitor to find an expression for the total energy stored in it.
Problem 3: Circuit with Capacitors in Series and Parallel
Consider the circuit shown to the right, where Vbat = 24.0 Volts, C1 = 1.00 μF, C2 = 2.00 μF, and C3 = 3.00 μF.
(a) Find the equivalent capacitance of the circuit, in micro-Farads (μF).
(b) Find the charge on each capacitor, in micro-Coulombs (μC).
Problem 4: Circuit with Multiple Capacitors
Consider the circuit shown below, where Vbat = 12.0 Volts and all capacitors have the same capacitance of 50.0 μF.
(a) Find the equivalent capacitance of the circuit, in μF.
(b) Find the charge on each capacitor, in μC.
Problem 5: Capacitor Charging and Discharging
Consider the circuit shown below. Initially, all capacitors are uncharged and the switch is open. The switch is then thrown to position A so that the battery can charge C1.
(a) Find the initial charge on capacitor 1 (Q1i), in micro-Coulombs.
(b) Find the initial energy stored in capacitor 1 (U1i), in Joules.
(c) The switch is then thrown to position B so that C1 can charge C2 and C3. Find the final charge on each capacitor (Q1f, Q2f, Q3f), in micro-Coulombs.
(d) Find the final energy stored in each capacitor (U1f, U2f, U3f), in Joules.
(e) Find the total final energy stored in all three capacitors, in Joules. Why is the answer different than your answer to part (b)?
Problem 6: Parallel Plate Capacitor with Dielectric
A capacitor is made up of two parallel plates of area 0.125 m2 separated by an insulating material of thickness 0.0150 mm and dielectric constant K = 20.0. The capacitor is connected to a 300.0 Volt battery to charge it.
(a) Find the capacitance (C) of the capacitor in Farads.
(b) Find the charge on the plates of the capacitor, in Coulombs.
(c) Find the charge on the surfaces of the dielectric (i.e., the induced charge), in Coulombs.
(d) Find the strength of the total electric field between the plates, in units of N/C.
(e) Find the strength of the electric field produced by the charges on the plates only, in units of N/C.
(f) Find the strength of the electric field produced by the charges on the dielectric only (i.e., the induced E-field), in units of N/C.
(g) Draw a diagram of the capacitor. Draw arrows that represent the two electric fields from parts (d) and (e). Also label the charges on the top and bottom plates of the capacitor, and the top and bottom of the dielectric.
Problem 7: Current and Resistance in a Platinum Wire
A cylindrical platinum wire is 0.0644 mm in diameter and 1.05 m in length. The wire carries a 1.05 A current. Assume there is one free electron (q = -1.602×10-19 C) per platinum atom. Avogadro’s number is 6.022×1023 atoms per mol. Platinum has a density of 21.5 g/cm3, a molar mass of 195 g/mol, and a resistivity of 10.6×10-8 Ωm.
(a) Find the current density (j), in units of A/m2.
(b) Find the number density of free electrons (n), in units of m-3.
(c) Find the drift velocity of electrons in the wire (vd), in units of m/s.
(d) Find the magnitude of the electric field in the wire, in units of N/C.
(e) Find the potential difference across the wire, in units of Volts.
Problem 8: Tungsten Filament in an Incandescent Light Bulb
An incandescent light bulb has a tungsten filament with an operating temperature of 2200.0 °C. The filament is 0.800 m long. At its operating temperature, the bulb is rated for a power output of 60.0 Watts when the voltage across it is 110.0 Volts. For tungsten, the resistivity is 5.60×10-8 Ωm at 20.0 °C, and the temperature coefficient of resistivity is 4.50×10-3 °C-1.
(a) Find the current through the filament at its operating temperature, in Amps.
(b) Find the resistance of the filament at its operating temperature, in Ohms.
(c) Find the cross-sectional area of the filament, in meters squared.
(d) Find the diameter of the filament, in millimeters.
(e) Find the resistance of the filament when it is cold (at 20.0 °C).
Problem 9: Basslink Project – DC Power Transmission
The Basslink Project has a cable that transmits DC power from mainland Australia to the island of Tasmania. It is rated to work with a 400.0×103 V power source while transmitting a total of 500.0×106 Watts of power (some of which is dissipated in the cable). A schematic of the system is shown below. The cable is made of copper (resistivity 1.68×10-8 Ωm), and is 295 km long with a cross-sectional area of 1200.0 mm2.
(a) Find the current in the cable, in Amps.
(b) Find the resistance of the cable, in Ω.
(c) Find the power loss in the cable, in Watts.
(d) Find the power used by the load in Tasmania, in Watts.
(e) Find the load resistance. This represents the effective resistance of all the devices the cable is powering.
Problem 10: Battery with Internal Resistance
A new battery has an EMF of ε = 1.60 Volts and an internal resistance r. When the new battery is connected to a load with resistance R = 3.00 Ω, there is a current I = 0.500 A flowing through it. The diagram below shows a schematic of the circuit.
(a) Find the internal resistance of the battery.
(b) Find the terminal voltage of the battery (VAB).
(c) What is the total power consumed by the load, in Watts?
(d) What is the total power consumed by the battery’s internal resistance, in Watts?
(e) As the battery gets old, it maintains its EMF of 1.60 Volts but its internal resistance increases. Now when the battery is connected to a load with resistance 3.00 Ω, the current is only 0.200 A. Repeat parts (a)-(d) for the old battery.
Problem 11: Circuit with Multiple Resistors
Consider the circuit shown on the right, where ε = 9.0 V, R1 = 10.0 Ω, R2 = 6.0 Ω, R3 = 8.0 Ω, R4 = 4.0 Ω, R5 = 5.0 Ω.
(a) Find the equivalent resistance of the circuit, in Ω.
(b) Find the current through each resistor, in Amps.
Problem 12: Circuit Analysis with Kirchhoff’s Rules
Consider the circuit shown below. The batteries have EMFs of ε1 = 10 V and ε2 = 2 V, and negligible internal resistance. The resistors in the circuit have the following resistances: R1 = 1 Ω, R2 = 2 Ω, and R3 = 3 Ω.
(a) Use Kirchhoff’s rules to write down three independent equations involving the currents shown on the diagram.
(b) Solve your system of equations to find the three unknown currents (I1, I2, I3).