Physics 227: Lecture 2 Coulomb’s Law, Superposition ... 227: Lecture 2 Coulomb’s Law, Superposition, Electric Fields, Field Lines • Lecture 1 review: • All your questions are

Physics 227: Lecture 2Coulomb’s Law, Superposition,

Electric Fields, Field Lines• Lecture 1 review:

• All your questions are answered on class web pages: Sakai: 01:750:227 or http://www.physics.rutgers.edu/ugrad/227

• Separate charges by rubbing (appropriate materials).

• There are two types of charges: + and -.

• Opposites signs attract, same sign repel.

• Conductors conduct, insulators do not.

• Charged objects ``polarize’’ uncharged insulators or conductors, leading to an attractive force.

Friday, September 9, 2011

http://www.physics.rutgers.edu/ugrad/227

http://www.physics.rutgers.edu/ugrad/227

Coulomb’s Law - force between two charges

q1

q2

F1 on 2 = k q1 q2 / r2

Consider two point charges q1 and q2 which have the same sign. The force is along the line connecting the charges.

F2 on 1 = k q1 q2 / r2

k = 8.99x109 N.m2/C2 or k = 1/4πε0 with

ε0 = 8.854x10-12 C2/N.m2

permittivity of free space

q1

q2

F1 on 2 = k q1 q2 / r2

If q1, q2 have opposite signs:F2 on 1 = k q1 q2 / r2

Note: until further notice, we are dealing with point charges, or with charge distributions that we assume are not significantly affected by the presence of other external charges. We ignore polarization effects such as we saw between a charged

rod and uncharged insulators or conductors in the first lecture.

Note the force is negative for charges of opposite signs - this indicates the

force is attractive.


q1

q2

More formally, with vectors, Coulomb’s law is:

In using this definition, you need to recall the definition of the direction of the unit vector as going from q1 to q2. Thus the force of q1 on q2 is in the direction from q1 to q2, if q1 and q2 have the same sign.

Coulomb’s Law - force between two chargesConsider two point charges q1 and q2 which have the same sign. The force is along the line connecting the charges.


Coulomb vs. Gravitational Force

Compare the strength of the Coulomb force to the strength of the gravitational force, between an electron and proton:

|FC| = k |q1 q2| / r2 = 9.0x109 n.m2/C2 (1.6x10-19 C)2 = 2.3x10-28 N / r2

|FG| = G m1 m2 / r2 = 6.7x10-11 n.m2/C2 (1.7x10-27 kg) (9.1x10-31 kg) = 1.0x10-67 N / r2

FC / FG ≈ 2x1039

Note: mearth ≈ 6.0x1024 kg so earth has ≈ 3.6x1051 protons + neutrons.

Both fall as 1/r2, so the distance does not matter.


Superposition

q3The total force on an object is the vector

sum of all the forces on the object.

q1

q2

F2 on 1 = k q1 q2 / r2

FTotal on 1

F3 on 1 = k q1 q3 / r2

Recall from introductory mechanics / statics that the sum of the internal forces within an object is 0, so we will not worry about those forces here.


Reminder - Vector Addition

Graphical addition:

Algebraic addition (2-dimensional example):

Slide tail of B to head of A, C goes from tail of A to head of B

A

BCBslid

Recall that a vector has a magnitude and direction. Its absolute position does not matter. If you slide it around, it is

still the same vector.


Simple Superposition Example

2 C -1 C

x=0, y=0

1 C

x=-1 m, y=0 x=1 m, y=0

What is the total Coulomb force on the -1 C charge at the origin?

q1 q2 q3


Iclicker:Superposition:

A. k

B. k

C. k

D. k

E. k

1 C

-1 C

x=0, y=0

1 C

x=0, y=1 m

x=1 m, y=0

What is the total Coulomb force on the -1 C charge at the origin from the other two 1 C charges?


Iclicker:Superposition:

A. k

B. k

C. k

D. k

E. k

1 C

-1 C

x=0, y=0

1 C

x=0, y=1 m

x=1 m, y=0

What is the total Coulomb force on the -1 C charge at the origin from the other two 1 C charges?


• We introduce the electric field from the force felt by a ``test’’ charge:

• Alternately, the electric field is the force felt by a unit charge, divided by 1 C (to get the units right).

• In doing this, we assume that the charges generating the force are fixed in place - they do not move around in response to the test / unit charge.

• The electric field should remind you of the gravitational field. For point masses / charges:

Electric Field

E is a ``vector field’’ - its magnitude and direction depend on position.

�g =�Fg

m= −GMe

r2r̂ → �E =

�FC

q=

kQ

r2r̂ =

Q

4π�0r2r̂


• For now it might just seem that the concept of electric fields is some math trick that does not make much difference - the electric field is some math abstraction rather than something that is real.

• We will later learn that the fields are real. For example:

• Energy is stored in electric and magnetic fields.

• Light is a traveling electric + magnetic field.

• Recall Einstein’s conception of gravity (General Relativity) geometrically, as curving space rather than being a force. Similarly, you can think a charge generates an electric field, like a mass generates a gravitational field, modifying space.

A note on Electric Fields


• Since forces are vectors, and add as vectors

• And FC = qE...

• ➭ Electric fields are vectors and add as vectors

• If we have charges qa, qb, qc, qd, qe, ... the force on charge qa is:

Superposition of Electric Fields

• Or Ftotal = qaE, with the electric fields adding:

When we are finding the force

on a charge qa, we only consider

fields generated by the other

charges, not by qa.


• Field lines show the electric field direction. By convention, field lines point from + to - charges. Field lines start on + charges and end on negative charges - or can start / stop at r = ∞.

• The density of field lines indicates the magnitude of the field. The field gets smaller away from a charge. A larger charge has more field lines that start / stop at it.

Field Lines

q1<0q2>0


Field Lines for Two Charges

• In the middle and right drawings, the two charges have the same number of field lines, so are equal in magnitude.• With two charges, the field lines curve. The E field at each point is tangent to the field line at that point - field lines cannot cross!• In the middle drawing, all the field lines start on the + charge and curve around to end on the - charge, except for two of the three horizontal ones which reach to r = ∞.


Field Lines for Two Charges• Why don’t we draw horizontal field lines pointing from each positive charge to the other?A. We have enough lines

already.

B. The fields from the two charges cancel between them.

C. The lines would point at each other.

D. The lines would not end at a charge or at r=∞.

E. Blue lines are not pretty. We need Rutgers red!


Field Lines for Two Charges• Why don’t we draw horizontal field lines pointing from each positive charge to the other?A. We have enough lines

already.

B. The fields from the two charges cancel between them.

C. The lines would point at each other.

D. The lines would not end at a charge or at r=∞.

E. Blue lines are not pretty. We need Rutgers red!

Field lines start/stop on charge or at r = ∞, have a unique magnitude & direction at each point in space, do not touch or cross


iClicker: Field Lines for Three ChargesWhat is the most you can determine about qtop, qmiddle and qbottom?

A. qt > qm > qb

B. qt > qm < qb

C. qt, qb > 0, qm < 0

D. qt, qb > 0, qm < 0 |qt| = |qb| > |qm|,

E. |qt| = |qb| > |qm|, qt, qb < 0, qm > 0


iClicker: Field Lines for Three ChargesWhat is the most you can determine about qtop, qmiddle and qbottom?

A. qt > qm > qb

B. qt > qm < qb

C. qt, qb > 0, qm < 0

D. qt, qb > 0, qm < 0 |qt| = |qb| > |qm|,

E. |qt| = |qb| > |qm|, qt, qb < 0, qm > 0


Thank you, andSee you next Monday


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Physics 227: Lecture 2 Coulomb’s Law, Superposition ... 227: Lecture 2 Coulomb’s Law, Superposition, Electric Fields, Field Lines • Lecture 1 review: • All your questions are