V. Electric Current and Resistance(qyPq ) 1. Definition: Current Electrostatics: charges are at rest. When charges (q

V. Electric Current and Resistance(電流與電阻)

1. Definition: Current Electrostatics: charges are at rest.

When charges (q) start to move, they generate a current (i), and

conservation of charge = conservation of current

2. Drift speed (漂移速度) and current Define the current density J:

Then and

The total charge within L is ,

where n is number density of charge carriers, A is the area of the cross section, and e is the charge of each carrier.

Assume that all carriers move along the wire with v_d, it takes for all these carriers within L to pass through any cross section.

Therefore or

3. Resistance and resistivity (電阻與電阻率)

or V = R i

Resister ( ): .

Resistivity (ρ) of the material: This is the “Ohm’s Law”.

NOTE:

4. Microscopic view of Ohm’s law

VI. Magnetic Fields (磁場)

1. Introduction

Definition: Magnetic field (B) and magnetic force

Recall the electric field is given by . Experiment showed that . And in a more detailed experiment:

Sample problem:

2. Magnetic field lines

3. Discovery of the electron

Thomspn’s procedure:

Deflection: y =1/2 a ∆t² = (check it!)

and thus

4. Hall effect (霍爾效應)

When an equilibrium is achieved (no more positive or negative charge is piling at the side), the electric potential difference V across d is stabilized (V = E d), and more importantly

FE = FB and thus ,

Because the drift speed is also given by ,

One can then derive he number density of carriers , where l = A/d, the thickness . This determines the magnitude and sign of carrier density.

5. Circulating charged particle

Question: What happens ? Answer: helical path.

6. Magnetic force on a current-carrying wire

Total charge passes through any cross section within a time t is

So

More exactly,

7. Magnetic dipole moment

Consider the torque on a current loop due to magnetic field

Torque: τ = r × F

(only F1 and F3 contribute; force F2 and F4 make no contribution because of cancellation) More exactly

τ = µ × B,

where the magnetic dipole moment (磁雙極矩)

µ = i A (Here A = ab.).

Recall the electric dipole moment (p) in a electric field, the corresponding torque is τ = p × E.