# Circuits

Magnetic field of a loop of wire: B = µ_{0} I / 2 R

Time rate of electric field on a capacitor: dE/dt = I / ( ε_{0} A )

(where A is the area of a capacitor plate)

Electric field between capacitor with a charge: E = Q / ( ε_{0} A )

For positive charges, F and E are opposite.

For negative charges, F and E are the same direction.

Potential energy between two charges: U = k q_{1} q_{2} / r_{12}^{2}

Θ = q_{enclosed} / ε_{0}

For an infinite line: q_{enclosed} = λ L and E = 2 k λ / r

For a sheet: E = ½ σ / ε_{0}

For two plates: E = - V / d (ΔV between plates, divided by distance between plates)

For a circle: E = k q / r^{2}

Cylindrical air coil: L (inductance) = µ_{0} N^{2} A / l

ΔV = I R = E d

ΔU = q ΔV for some test charge

Q = C V = ε_{0} A E

W = ΔU = - q ΔV

ΔV = ΔU / q = k q / r

℘ = I^{2} R = I ΔV = ΔV^{2} / R = I Ε

Ε = I R + I r (way to find epsilon when an external resistor is in the circuit)

R is external resistance, r is internal battery resistance

ΔV = Ε - I r

I = Ε / ( R + r)

Epsilon is EMF, the Delta V when no resistor is in place if you will, and delta V is the voltage difference when the resistor is connected.

Series | Parallel |

Diff ΔV | Same ΔV |

Same Q | Different Q |

C_{total}^{-1} = C_{1}^{-1} + C_{2}^{-1} |
C_{total} = C_{1} + C_{2} |

V_{total} = V_{1} + V_{2} |
V_{1} = V_{2} |

R_{total} = R_{1} + R_{2} |
R_{total}^{-1} = R_{1}^{-1} + R_{2}^{-1} |

I_{total} = I_{1} = I_{2} |
I_{1} = I_{total} ( R_{total} - R_{1} ) / R_{total} |