Wednesday, February 15, 2006

Electricity and Magnetism Tutorial

Important Points from Lectures

a) Positive and Negative Charge

attraction and repulsion

(+) --> <-- (-), <-- (+) (+) -->, <-- (-) (-) -->

b) Electroscope

Charged rod touches neutral electroscope. Electroscope becomes charged. Charge on electroscope redistributes. "Leaves" have same charge, so they repel each other. At some separation, electrostatic force equals gravity so it comes to rest. Height increases with strength of force so it is used to measure charge.

c) Conservation of charge


d) Coulomb's Law


magnitude of force between two charges. Direction of the force is along line joining charges, in directions given by a (above).

e) Superposition

Force on a charge due to a collection of other charges is the sum of the forces due to every individual other charge considered separately.


f) Electric Field


g) Gauss' Law

Flux through a closed surface is proportional to the charge contained within the surface.

h) Conductors

No field inside a conductor, field at surface always perpendicular to surface.

i) Field lines

Point in direction of field, density indicates strength

j) Electric potential

Work required to move a charge from one point to another is equal to the increase in potential energy. Potential (φ) is energy per unit charge.

Sample Problems

1) What is the magnitude of the force a 30µC charge experiences when it is 40cm away from a 3mC charge?
    F=kQ1Q2/r2 = 5000N
2) A 1m square has +/- 12µC (= +/-Q) on the corners. What is the force experienced by each charge (magnitude and direction)

First, let's find the field at the top left corner - we'll call this point p.

Use superposition to find the fields at p due to the other three corners (let's call them 1, 2 and 3 respectively clockwise around the square from point p) separately:

1) |E1|=kQ/12. As Q1 is a positive charge, the field must point away from it - in the negative x direction.

2) |E2|=kQ/(√2)2. Q2 is negative, so the field points towards it - this vector is 45o below the x-axis: (1/√2, -1/√2)

3) |E3|=kQ/12. As Q3 is a positive charge, the field must point away from it - in the positive y direction.

Applying superposition, the total field is:

    E = kQ*[(-1,0)+(0,1)+1/2(1/√2,-1/√2)]=kQ*(1/2√2-1,1-1/2√2)
As the charge at p is "-Q"
    F = -QE = -kQ2*(1/2√2-1,1-1/2√2) = -kQ2(2√2-1)/(2√2)*(-1,1)


aimed at the center of the square.

What about the force on the other charges? Symmetry tells us immediately that each of them has the exact same force (magnitude) all directed towards the center.

More Gauss' Law Examples

What is the electric field near an infinitely long uniform line of charge?
    1) Let's define λ as the charge per unit length (units: coulomb/meter).

    2) Symmetry tells us that the field must everywhere be radiating directly outwards from the line and have the same magnitude at all points at a given distance from the line. (This can be shown by considering a rotation of the line about its axis and the mirror inversion of the wire along its length)

    3) To use Gauss' Law, we need to construct a surface- the obvious choice is a cylinder L meters long with a radius r, centered on the line.

    Then, the field is the same magnitude every where on the curved face and also perpendicular to the face, but parallel to the end faces.

Therefore: Total Flux = 2πrLE(r) = Q(L)/ε0, where Q(L) is the charge contained in L meters of line.

    2πrLE(r) = λL/ε0

    E(r) = λ/2πrε0

What about a horizontal, infinite plane of charge? Follow a similar process. σ surface charge density (C/m2) Symmetry: Field lines are vertical, no horizontal variation in field strength, Field strength h above plane, E(H), is same a h below plane (to see this, think about rotating, translating and flipping over the plane)

Construct a "pillbox" for surface:

Horizontal cross section A, extends height h above and below plane.

Flux through surface: 2AE(h)=Q(A)/ε0=Aλ/ε0


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