The magnetic force law (Lorentz law)

The magnitude of the force is

the B field x the perpendicular velocity x charge

Or

The velocity x the perpendicular B field x charge

Use right hand rule, but be careful … notice sign of charge!

BvqFvvv

×=

θsinvBqF =v

Another way to take a cross product

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vA ×

v B =

√ i √ j √ k

Ax Ay Az

Bx By Bz

We discussed the flux of the magnetic field last time. Through any closed surface, it’s zero.

∫ =⋅ 0AdBrv

No “sources” (origins) for magnetic field lines.

We will see in the coming month that magnetic flux through open surfaces IS important. It’s calculated the same way we calculated flux of the electric field…

A B field of 10 Tesla (T) is pointing north and up (30° above horizontal.)

What is the flux of the B field through the desk, A= 2 m2 ?Recall cos(30°) = √3/2

cos(60°) = 1/2

A] 5 Tm2

B] 5√3 Tm2

C] 10 Tm2

D] 10√3 Tm2

E] 20 Tm2

As the magnetic force is always perpendicular to velocity, a uniform B field cannot cause particles to

speed up or slow down.

It just makes them go in circles. (OR HELICES)

Centripetal force = Fmag

Cyclotron frequency

Part a. Use A for i), B for ii) etc.

Radius is proportional to v. Ans. B (ii)

Part b. Use A for i), B for ii) etc.

Because r is proportional to v, the period is unchanged. Ans. A (i)

Magnetic forces on current carrying wires.

Current means charges in motion. The field exerts a force on the moving charge carriers. They transfer that force to the lattice through collisions.

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vF = I

v l ×

r B wire

v F = q

v v ×

r B charge

The unit of the magnetic field B (the Tesla)

A] is the same as the electric field times a velocity

B] is the same as the electric field divided by a velocity

C] cannot be expressed as either of these

(The electric field is V/m.)

velocity

Use A for i), B for ii) etc.

Same speed. Ans C (iii)

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vF = I

v l ×

r B wire

v F = q

v v ×

r B charge

A curious point: since magnetic forces are perpendicular to velocity,

they do no work on charges.

But they do work on wires!

How can this be? (I’m not going to tell you the answer…life should have some mysteries, eh?)

The electric potential isA] higher at AB] higher at BC] the same in both places

If the current direction were the same, but electrons carried the current, where would the electric potential be higher?

Hall Effect -- How to determine the sign of charge carriers

What is the sign of the charge carriers in this conductor?

A] +

B] -

C] Can’t tell

What is the drift velocity of the negative charge carriers?

A] 0B] 10-3 m/sC] 10-2 m/sD] 10-1 m/sE] 1 m/s

Why do compass needles align with B?Why do opposite poles attract?Why do magnets attract iron?

BU

AI

B

vv

vv

vvv

⋅−=

=

×=

μμ

μτMagnetic Torque on a current loop

Magnetic Moment

Potential energy of a current loop

Sources of Magnetic Fields - Moving Charges

Biot-Savart Law

1/r2 dependence in a given directionNo B field front or backB field “curls” around charge trajectoryμ0 is a constant, just like 0 for E fields. “Permeability of space”

Sources of Magnetic Fields - Moving Charges

Biot-Savart Law

A charge moves in a circle at constant speed.At the time shown, where is B=0?Or choose E: at more than one point shown

Sources of Magnetic Fields - Moving Charges

Biot-Savart Law

A charge moves in a circle at constant speed.If one wishes to calculate the B field at point D, what is ?

A) A unit vector upward B) a unit vector downward

r)

Sources of Magnetic Fields - Moving Charges

Biot-Savart Law

A + charge moves in a circle at constant speed.What is the direction of the B field at D?

A] left B] into page C] out of page D] down E] up

Sources of Magnetic Fields - Moving Charges

Biot-Savart Law

A + charge moves in a circle at constant speed.What is the direction of the E field at D?

A] left B] into page C] out of page D] down E] up

Sources of Magnetic Fields - Moving Charges

Biot-Savart Law

A + charge moves in a circle at constant speed.What is the direction of the B field at point B?

A] out of page B] out of page and up C] out of page and up and right

Sources of Magnetic Fields - Moving Charges

Biot-Savart Law

What is the ratio of the B field strength at C to that at D?

A] BC/BD=4 B] BC/BD=2 C] BC/BD=1 D] BC/BD=1/4

Sources of Magnetic Fields - Moving Charges

Biot-Savart Law

The “pièce de résistance!” What is the ratio of the B field strength at C to the strength at B?

A] BC/BB=4 B] BC/BB=2√2 C] BC/BB=2 D] BC/BB= BC/BD=√2

Just as with force laws,

I l q v

This “version” of the right hand rule gives the same result as Try it.

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dv l ×

ˆ r

What is the contribution of the straight sections of the wire to the magnetic field in the center of the semicircle?

A] Each contributes 0

B] They both have contributions that are opposite, and so add to 0

C] Each contributes

D] infinite

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μ0I

2πR

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−∞

€

+∞

Now let’s work out the contribution from the semicircle.