Phys 115 midterm 1

Coulomb’s Law (force magnitude)

F = k|q₁||q₂| / r²

• F = force between the charges (N)

• k = 8.99×10⁹ N·m²/C²

• q₁, q₂ = the two point charges (C)

• r = distance between them (m)

Electric field (definition)

E = F / q₀

• E = electric field (N/C)

• F = force on the test charge (N)

• q₀ = test charge (C)

Force on a charge in a field

F = qE

• F = force on the charge (N)

• q = the charge (C)

• E = electric field at its location (N/C)

Field of a point charge

E = k|Q| / r²

• Q = the source charge (C)

• r = distance from Q (m)

Field inside a parallel-plate capacitor

E = Q / (ε₀A) = σ / ε₀

• Q = charge on one plate (C)

• A = plate area (m²)

• ε₀ = 8.85×10⁻¹² C²/N·m²

• σ = surface charge density (C/m²)

• Points from + to −.

Surface charge density

σ = Q / A

• Q = charge (C)

• A = area (m²)

Potential difference (from work)

ΔV = −W_elec / q₀ = ΔU / q₀

• ΔV = potential difference (V = J/C)

• W_elec = work done by the electric force (J)

• ΔU = change in electric potential energy (J)

• q₀ = charge (C)

Work and potential energy

W_elec = −ΔU_elec

• W_elec = work done by the electric field (J)

• ΔU_elec = change in electric potential energy (J)

Energy conservation (charge + field system)

ΔK = −ΔU_elec

• ΔK = change in kinetic energy (J)

• ΔU_elec = change in electric potential energy (J)

Potential of a point charge

V = kq / r

• q = source charge (C)

• r = distance from the charge (m)

• V = 0 at r = ∞.

Potential energy of two point charges

U = kq₀q / r

• q₀, q = the two charges (C)

• r = separation (m)

Field and potential (uniform field)

E = −ΔV / Δx

• E = electric field (V/m = N/C)

• ΔV = potential difference (V)

• Δx = distance along the field (m)

Dipole moment

p = qd

• p = dipole moment vector (C·m)

• q = magnitude of each charge (C)

• d = vector from the − to the + charge (m)

• p points from − to +.

ECG voltage

ΔV_ECG = V₊ − V₋

• V₊ = potential at the positive terminal (left arm)

• V₋ = potential at the negative terminal (right arm)

Capacitance (definition)

C = Q / ΔV_C

• C = capacitance (F)

• Q = charge on one plate (C)

• ΔV_C = voltage across the plates (V)

Parallel-plate capacitance

C₀ = ε₀A / d

• ε₀ = 8.85×10⁻¹² C²/N·m²

• A = plate area (m²)

• d = plate separation (m)

Dielectric: capacitance

C = κC₀

• κ = dielectric constant (unitless, κ ≥ 1)

• C₀ = capacitance without the dielectric (F)

Dielectric: field

E = E₀ / κ

• E₀ = field without the dielectric (N/C)

• κ = dielectric constant

Energy stored in a capacitor

U = ½QV = ½CV² = Q²/2C

• U = stored energy (J)

• Q = charge (C)

• V = voltage across the capacitor (V)

• C = capacitance (F)

Energy density of a field

uₑ = ½κε₀E²

• uₑ = energy per unit volume (J/m³)

• κ = dielectric constant

• ε₀ = permittivity of free space

• E = electric field (N/C)

Current

I = ΔQ / Δt

• I = current (A = C/s)

• ΔQ = charge passing a point (C)

• Δt = time interval (s)

Ohm’s Law

ΔV = IR

• ΔV = voltage across the resistor (V)

• I = current through it (A)

• R = resistance (Ω)

Resistance of a wire

R = ρL / A

• ρ = resistivity of the material (Ω·m)

• L = length (m)

• A = cross-sectional area (m²), which is πr² for a round wire

Total current in a circuit

I_total = V_battery / R_total

• V_battery = battery voltage (V)

• R_total = equivalent resistance of the circuit (Ω)

Resistors in series

R_eq = R₁ + R₂ + … + R_N

• Each Rᵢ = an individual resistance (Ω)

• Same current through all of them.

Resistors in parallel

1/R_eq = 1/R₁ + 1/R₂ + … + 1/R_N

• Same voltage across each. Remember to take the reciprocal at the end.

Kirchhoff’s Junction Law

ΣI_in = ΣI_out

• I_in = currents entering a junction (A)

• I_out = currents leaving it (A)

• Based on conservation of charge.

Kirchhoff’s Loop Law

ΔV_loop = ΣΔVᵢ = 0

• ΔVᵢ = potential change across each element in a closed loop (V)

• Based on conservation of energy.

RC time constant

τ = RC

• τ = time constant (s)

• R = resistance (Ω)

• C = capacitance (F)

RC charging: capacitor voltage

ΔV_C = ℰ(1 − e^(−t/τ))

• ΔV_C = capacitor voltage at time t (V)

• ℰ = battery voltage, V_battery (V)

• t = time since the switch closed (s)

• τ = RC

RC charging: current

I = I₀e^(−t/RC)

• I = current at time t (A)

• I₀ = initial current = ℰ/R (A)

• R, C = resistance (Ω) and capacitance (F)

• t = time (s)

RC discharging: capacitor voltage

ΔV_C = (ΔV_C)₀ e^(−t/RC)

• (ΔV_C)₀ = capacitor voltage at t = 0 (V)

• R, C = resistance (Ω) and capacitance (F)

• t = time (s)

What does inserting a dielectric do to capacitence?

Raises it