Practice Problems for Potential and
Capacitance
1. Consider the distribution of point charges shown below. The coordinate axes are marked in centimeters.
a) Find the potential at the origin for this distribution of charges.
b) Calculate the electric potential on the y-axis at y=4 cm.
c) How much work much be done to move the above charge qo from the origin to y=4 cm?
2. Consider the Bohr model of the hydrogen atom: a proton (charge qp=1.6x10-19 C) lies stationary in the middle of the atom and a single electron orbits in a circle around the proton. The Bohr model predicts that the electron can only occupy set orbits with specific radii. The radius of the first orbit is r1=0.53x10-10 m and the second orbit is r2=2.12x10-10 m.
a) Calculate the point charge potential due the proton at the position of the first Bohr radius r1.
b) Calculate the point charge potential due the proton at the position of the second Bohr radius r2.
c) Ignoring the electron, what is the shape of the equipotential surfaces surounding the proton?
d) Find the electric potential difference DV between the two orbits.
e) How much energy is needed to move an electron from the first Bohr radius r1 to the second r2?
3. A capacitor is made by placing a 0.5 mm sheet of polyethelene (permittivity 2.03x10-11 C2/Nm2) between two sheets of aluminum foil with dimensions 0.05 m X 2.0 m (total area of 0.1 m2).
a) Calculate the capacitance of this configuration of conductors.
b) Calculate the strength of the electric field between the capacitor plates. Remember there is a dielectric between the foil sheets.
c) If the capacitor is connected to a 9 V battery, then how much charge will accumulate on each of the plates?
d) How much energy is stored by the capacitor if it is hooked up to a 12 V car battery?
e) How much energy is stored by the capacitor if a charge of 5x10-5 C accumulates on each of the aluminum sheets.
4. A vacuum-gap parallel plate capacitor contains plates which are 1 mm apart giving the capacitor a total capacitance of 0.3 mF.
a) If there is a potential difference of 10 V across the plates, then what is the electric charge on each plate?
b) Calculate the strength of the electric field inside the capacitor plates for a).
c) Briefly explain what will happen if a dielectric material is placed between the plates of the capacitor.
d) How much energy is released when the terminals connected to the two plates of the capacitor described in a) are shorted by a conducting rod (like a screwdriver)?
5. Consider the capacitive circuit shown below:
a) Calculate the total equivalent capacitance for this circuit.
b) What is the total amount of charge provided by the voltage source when the capacitors are fully charged?
c) How much energy is provided by the battery to the circuit?
6. Consider the below capacitive circuit.
24V
a) Find the equivalent capacitance for this circuit.
b) How much total charge is provided to the circuit by the battery?
c) What is the charge which accumulates on the 8 mF capacitor?
d) Calculate the potential difference across the 8 mF capacitor.
e) How much energy is stored by the 6 mF capacitor which is in series with the 8mF cap?
f) How much energy is stored by the entire circuit? That is, how much energy does the battery provide?
7. Three point charges are placed as shown in the coordinate system below.
a) Calculate the electric potential at the origin by assuming the potential infinitely far away is zero.
b) Calculate the potential at (x=5m, y=3m).
c) How much energy is required to move a charge of 1x10-6 C from the origin to (x=5m,y=3m)?
8. Two sheets of metal with a length of 0.5 m and width 0.2m are placed a distance of 0.0001m apart. All of the air is pumped out from between the two sheets. Also, the arrangement of conductors is connected to a 20 V battery. Answer the following questions for the arrangement of conductors described.
a) Calculate the capacitance for the conductors described.
b) What is the strength of electric field between the plates?
c) How much energy is stored by this parallel plate capacitor?
d) What is the charge on each of the two metal sheets?
e) A dielectric with a permittivity of 5x10-11 C2/Nm2 is placed between the two conductors. Calculate the new electric field between the conductors.
f) How much energy is stored in the capacitor with the dielectric between the plates?