School

Harvard UniversityDepartment

Physical SciencesCourse Code

Physical Sciences 3Professor

Roxanne GuenetteStudy Guide

FinalThis

**preview**shows pages 1-3. to view the full**10 pages of the document.**Physical Sciences 3: Lecture 1c

February 5, 2019

1

Current and Resistance

• Key points from the pre-reading:

An insulator has essentially no charge that is “free to move.” Although a potential

gradient will exert a force on charges in an insulator, the charges remain bound and do

not flow. A conductor contains charge that is free to move. If there is a potential

gradient in a conductor, these charges will feel a force and will flow in response.

A conductor at equilibrium will have the same potential everywhere. An empty cavity

inside a conductor at equilibrium will also have the same uniform potential. Thus, there

will be no potential gradient inside a conductor at equilibrium.

If a potential difference is applied across a conductor, charge will flow. The flow of

charge is called current. The current i that results can be given by any of the following

relationships:

From resistance (Ohm’s Law):

From resistivity

ρ

or conductivity

σ

: or

• For ions in solution, the total conductivity is the sum of the conductivity due to each

species of ion in the solution. (The solution must overall be electrically neutral.) The

conductivity of a particular species of ion is modeled simply as:

: c is the concentration, q is the charge, and f is the drag coefficient of the ion

•Learning objectives: After this lecture, you will be able to…

1. Describe the difference between a conductor and an insulator.

2. Explain why any conductor at equilibrium will have the same electric potential

everywhere on its surface and inside it.

3. Derive a simple model to show how the electric current carried by ions in a solution is

related to:

• the charge of the ion, q

• the drag coefficient of the ion, f (this is viscous drag like Stokes’s Law)

• the concentration of ions, c

• the potential gradient across the solution, ΔV/L

• the cross-sectional area of the solution, A

4. Describe the behavior of conductors in terms of conductivity or resistivity.

5. Derive Ohm’s Law that relates current, voltage, and resistance

i=ΔV

R

i=ΔV

ρ

A

L

i=ΔV

σ

A

L

σ

=cq2

f

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

Physical Sciences 3

February 5, 2019

Activity 1: Continuous Charge Distributions: Ring of Charge

• Often, we will want to model a continuous distribution of charge. In this case, we’ll need

to replace the sum with an integral. How can we express the superposition principle?

• So what will be the tiny (infinitesimal) potential created by a tiny bit of charge?

• Usually, we’ll express the tiny bit of charge in terms of a density of charge:

For a linear density (charge per unit length):

For an area density (charge per unit area):

For a volume density (charge per unit volume):

• What the potential on the axis of a thin ring of charge, with linear charge density

λ

?

• What if you’re very far away from the ring (i.e. x >> R)?

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

Physical Sciences 3: Lecture 1c

February 5, 2019

2

Continuous Charge Distributions: Disk of Charge

• Now let’s consider a uniformly-charged disk with area charge density

σ

. What is the

potential at any point on the axis of the disk?

• What about the limit when you’re very close to the disk (i.e. x << R0)?

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