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Washington University in St. Louis

Physics

Physics 198

Seidel

Spring

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Physics 198 Exam I: E1-E10
I. E1- Electric Fields
βββ 1 π1 2 1 9 2 2
A. |πΉ π = 4ππ0 π2 , where π = 4ππ0 = 8.99 x 10 Nm /C
1. Opposite signs Β» attractive
2. Same signs Β» repulsive
βββ β
B. |πΉ π = ππΈ
1. Generally electric field lines point away from + charges, towards negative charges
1 π
2. πΈ = 4ππ π2 πΜ
0
3. Superposition principle: total electric field is the vector sum of each individual field,
unaffected by presence of other charges
II. E2- Charge Distributions
A. Dipoles
1. πΈ = 1 π π = 1 πβπβ
2ππ0π3 2ππ0π3
a. Where s is the separation between charges, r = distance from center of dipole
b. Where p ie the dipole moment vector, direction of positive particle with respect
to negative particle βπββ = ππ + Μπ
B. Electric Polarization
β β 2πΌ|πΈ| π 2
1. πβπββ = πΌπΈ, πΈ = (4ππ0) π 5(higher ο‘ is higher polarizability)
C. Calculating Electric Fields
1. Divide distribution into small bits to model as particles
2. Determine each bitβs charge
3. Calculate the field vector that each point contributes at P
4. Sum the field vectors over all bits
D. Other Charge Distributions
1. Charge of an Infinite Plane
β π
a. πΈ = 2π0πΜ (perpendicular to source plane at P, w/ constant charge per unit area ο³)
2. Spherical Shell
a. Inside the shell, πΈ = 0
1 π
b. Outside the shell: πΈ = 2πΜ
4ππ 0
3. Infinite Line
β 1 π
a. πΈ = 2ππ0π πΜ
4. Along the axis of a ring with uniform charge Q
a. πΈ = ππ§π (fields cancel in x and y directions b/c of circle symmetry)
π (π§ +π
)3/2
III. E3- Electric Potential
A. Electric Potential Energy
π π π
π = ( 1 + 2 + β―)
π 4ππ 0 |π1 π2|
B. Electric Potential ππ(π₯,π¦,π§) 1 π
1. π π₯,π¦,π§ =) π = 4ππ |π|
βππ
2. In a uniform field: βπ = = βπΈβπ,
π
a. For any field: βπ = β πΈ β«πβ
b. πΈ π₯,π¦,π§ = ββ π
3. Shell Theorem
1 π
a. Inside a shell, potential is constant: π =
1 π 4ππ0π
b. Outside a shell: π =
4ππ0|π|
IV. E4- Static Equilibrium
A. Static Equilibrium
1. Charge is at rest within conductorβs boundaries, πΈ = 0 inside of surface
a. Potential is the same at all points inside an isolated conducting region (βο¦ =0)
b. πΈ at all points on conductorβs surface must point perpendicular to that surface
c. Conducting interior of a conducting object must be electrically neutral in static
eqβb
2. Can isolate an object form an external field using a conducting container (Farraday
cage), protrusions on a surface have a higher charge density
B. Parallel Capacitor
π π΄
1. Capacitance, πΆ = βπ = π 0π
a. Large C holds more charge
b. C depends only on size, shape, arrangement of plates
c. A is the area of plates, s is the distance between the plates
π
2. Between 2 plates, πΈ = (where ο³ = Q/A) and βπ = β|πΈ|π β
2 π0
a. |πΉ| = π
20 π΄
1 β 2
b. Electric field density Β΅π= π2|πΈ0
1 2
c. Energy stored in capacitor = U = πΆ|2π|
V. E5- Current
A. Current Density (π½)
1. π½ = ππ£ = πππ = ππ( )π = ππ ( )π = ο³ πΈ2 πΈ β
π π
π π πΆ
ππΈβ
a. Drift velocity π£ π π π, Drude theory uses π estimation
b. n is the number density of carriers, π =-1ime between collisions
c. ο³ iπΆ the conductivity in units of (Ξ©π)
B. Flux
βπ
1. πΉππ’π₯ = = π π£ |π΄| = π π£ |π΄|πππ ο± = π½ β’ π΄
βπ
a. Total flux through a surface, π = β« π½ β’ ππ΄
π π
2. Current (I) a. πΌ = β«π π½ β’ ππ΄, expresses the rate at which charge flows through a given surface
b. For a uniform wire, I = JA (A = cross sectional wire area)
c. Flux is positive if charges move forward, negative if backwards through a surface

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