7 - Electrostatics, Gauss's Law, Electric Potential

Question 1

What is Coulomb’s Law?

Answer 1

A law stating that like charges repel and opposite charges attract, with a force proportional to the product of the charges and inversely proportional to the square of the distance between them.

F = (kq_1q_2)/r^2

Question 2

What is an electric field?

Answer 2

A region around a charged particle or object within which a force would be exerted on other charged particles or objects.

Question 3

What is electric flux?

Answer 3

Electric flux is the measure of the electric field through a given surface, although an electric field in itself cannot flow.

Φ_E = E * S = ES * cos θ

Question 4

What is Gauss’s Law?

Answer 4

The total of the electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity.

Φ_E = Q / ε_0 = ∮E * dA

Question 5

What is electric potential?

Answer 5

Electric potential energy is the energy that is needed to move a charge against an electric field.

V = k * q/r

Question 6

What is a capacitor?

Answer 6

Capacitors store energy by holding apart pairs of opposite charges. Since a positive charge and a negative charge attract each other and naturally want to come together, when they are holding a fixed distance apart, their mutual attraction stores potential energy that is released if they are reunited.

The simplest design for a capacitor is a parallel plate, which consists of two metal plates with a gap between them: electrons are placed onto one plate (the negative plate), while an equal amount of electrons are removed from the other plate (the positive plate).

Any two conductors separated by an insulator form a capacitor.

Question 7

What is the formula for capacitance?

Answer 7

C = Q / V_ab = ε_0 * A/d

The capacitance C is a property of the device itself, it doesn’t depend on the charge stored.

Question 8

What is a dielectric?

Answer 8

An insulating material or a very poor conductor of electric current. When dielectrics are placed in an electric field, practically no current flows in them because, unlike metals, they have no loosely bound, or free, electrons that may drift through the material. Instead, electric polarization occurs.

The positive charges within the dielectric are displaced minutely in the direction of the electric field, and the negative charges are displaced minutely in the direction opposite to the electric field. This slight separation of charge, or polarization, reduces the electric field within the dielectric.

k = ε / ε_0

k = dielectric constant
ε = permittivity of the substance
ε_0 = permittivity of free space/vacuum

Question 9

How does a dielectric affect the capacitance?

Answer 9

A capacitor has increased capacitance when a dielectric is added.

C = ε_0 * k * A/d

Question 10

What are the units of a dielectric?

Answer 10

Volts per metre (V/m)

Question 11

What happens to the voltage when a capacitor’s plates are separated?

Answer 11

The voltage increases when the plate separation is increased. It is most likely because of the distance. The charge has to cover a bigger distance if the distance is larger. This means that there will be a decrease in the total capacitance.

Question 12

What happens to the voltage when a dielectric is inserted between a capacitor’s plates?

Answer 12

Introducing a dielectric into a capacitor decreases the electric field, which decreases the voltage, which increases the capacitance.

Question 13

What is another word for a capacitor?

Answer 13

Condenser

Question 14

What is a qubit?

Answer 14

Another term for a quantum bit.

Question 15

What are the units of electric flux?

Answer 15

Newton metres^2 per Coulomb

Question 16

What is the electric field of a static conductor?

Answer 16

Zero

Question 17

Where is excess charge in the conductor located?

Answer 17

On the surface

Question 18

What is a Gaussian surface?

Answer 18

A Gaussian surface (sometimes abbreviated as G.S.) is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, the electric field, or magnetic field.

Question 19

Why is the electric field zero outside a capacitor?

Answer 19

The electric field due to a plate of the capacitor is independent of the distance from it (it’s uniform) provided it’s not infinite. So if the finite identical plates have uniform charge density, away from the edges outside the capacitor the field should be 0.

Question 20

What is charge density?

Answer 20

Charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter, at any point in a volume.

σ = Q/A

Question 21

What is the change in electric potential?

Answer 21

V_ab = V_a - V_b = a∫b E~ * dl~

Question 22

How much energy can be stored in a capacitor?

Answer 22

Energy stored is the work done in moving charge from -Q to +Q plates.

dW = V dq = q/C dq

W = Q^2 / 2C

Question 23

Can we store any amount of energy in a capacitor?

Answer 23

No. Eventually, the electric field is large enough to tear electrons of the atoms in the material between the plates.

U = Q^2 / 2C = 1/2CV^2 = 1/2QV

Question 24

What is the capacitance of two capacitors in parallel?

Answer 24

C_eq = (Q1 +Q2)/V = C1 + C2

Question 25

What are the expressions for two capacitors in a series?

Answer 25

V = V1+V2

V1 = Q/C1
V2 = Q/C2

V = Q(1/C1 + 1/C2)

C_eq = Q/V = 1/(1/C1 + 1/C2)

Question 26

A parallel plate capacitor stores a charge of +Q on one plate and -Q on the other plate at a potential difference of ΔV. If the voltage applied by a battery is doubled to 2ΔV, what happens?

Answer 26

The capacitance remains the same and the charge doubles

Question 27

The plates of a parallel plate capacitor were cut so that their area was reduced to half their initial value. At the same time, the distance separating the plates was increased to double its initial value. What happens to the capacitance?

Answer 27

The final capacitance is a quarter of the initial capacitance

Question 28

A battery is used to charge a parallel-plate capacitor. A dielectric slab is then inserted between the plates of the capacitor while the battery remains connected. What happens to the charge?

Answer 28

The charge on the plates is increased.

Question 29

Three capacitors of identical value, C, are connected with two in series and one in parallel. What is the equivalent capacitance?

Answer 29

C_eq = 1/(1/C + 1/2*C)

= 3C/2

Question 30

A parallel plate capacitor is fully charged by a battery. A dielectric slab is then inserted between the plates of the capacitor after the battery is disconnected. What change would you expect in the stored energy of the capacitor?

Answer 30

The energy would decrease

Question 31

Capacitors rarely consist of only two plates but are usually made up of multiple plates. If there are M plates (M even: 2, 4, …), each with area A and separation d, derive an expression for the capacitance C of such a capacitor. You may ignore edge effects.

Answer 31

C = (ε0A/d)(M-1)

Question 32

One conductive plate has a charge Q and another plate has a charge +1.2Q. The plates are separated from each other by a distance d to form an air-filled parallel plane with area A.

a. ) What is the capacitance C of the capacitor?
b. ) What is the electric field between the plates of the capacitor?
c. ) What is the potential difference between the plates of the capacitor?