AP Physics B & C –Electric Circuits –Equations to be Remembered

You must remember the following equations to make you strong in answering multiple choice questions involving electric circuits:

(1) Electric current, I = nAvq where n is the number density (number per unit volume) of mobile charge carriers, A is the area of cross section of the conductor, v is the drift velocity of the charge carriers and q is the charge on each carrier.

Since the charge carriers in a conductor are electrons of charge e, the above expression becomes

I = nAve

(2) Resistivity ρ = RA/L where R, A and L are respectively the resistance, area of cross section and the length of the conductor.

(3) If a wire of resistance R is stretched so that its length becomes n times its original length, its area of cross section becomes (1/n) times its original area of cross section. The resistance of the stretched wire therefore becomes n²R.

[The stretch can be described also in terms of the radius of the wire. Thus if a wire of resistance R is stretched so that its radius becomes (1/n) times the original radius, its area of cross section becomes (1/n²) times the original area of cross section and its length becomes n² times the original length. The resistance of the wire therefore becomes n⁴R].

(4) Conductivity s is the reciprocal of resistivity: s = 1/ρ

(5) Effective value (R) of resistances in series: R =R₁ +R₂ +R₃ +R₄ +….etc.

(6) Effective value (R) of resistances in parallel is given by the reciprocal relation,

1/R = 1/R₁ + 1/R₂ + 1/R₃ +… etc.

(7) (a) The equivalent emf of a series combination of n cells is the sum of their individual emf’s:

Ɛ = Ɛ₁ + Ɛ₂ + Ɛ₃ + Ɛ₄ +……etc

(b) The equivalent internal resistance of a series combination of n cells is the sum of their internal resistances:

r = r₁ + r₂ + r₃ + r₄ +……etc

(8) If there an n cells of emf Ɛ₁, Ɛ₂, Ɛ₃, …… Ɛ_n, and of internal resistances r₁, r₂ , r₃, r₄……. r_n respectively, connected in parallel, the combination is equivalent to a single cell of emf Ɛ_eq and internal resistance r_eq, such that

1/r_eq = 1/r₁ +1/r₂ +1/r₃ +……..+1/r_n and

Ɛ_eq /r_eq = Ɛ₁/r₁ + Ɛ₂/r₂ + Ɛ₃/r₃ +…….+Ɛ_n /r_n

(9) Kirchhoff’s rules:

(i) Junction rule: At any junction, the sum of the currents entering the junction is equal to the sum of currents leaving the junction.

(ii) Loop rule: The algebraic sum of changes in potential around any closed loop involving resistors and cells is zero.

(10) Power in a D.C. circuit = VI = I²R = V²/R where V is the voltage, I is the current and R is the resistance.

(11) If devices consuming powers P₁, P₂, P₃, P₄ ….etc. (at the same supply voltage V) are connected in parallel across the supply voltage V, the total power consumed by them is given by

P = P₁+P₂+ P₃+ P₄ +….etc.

(12) If devices consuming powers P₁, P₂, P₃, P₄ ….etc. (at the same supply voltage V) are connected in series and the series combination is connected across the supply voltage V, the total power consumed by them is given by the reciprocal relation,

1/P = 1/P₁ +1/P₂ +1/P₃ +1/P₄ +….etc.

(13) A cell will transfer maximum power to a load if the internal resistance of the cell is equal to the resistance of the load.

Transients in RC circuits are additionally included for the AP Physics C Examination. AP Physics C aspirants should therefore remember the following points also:

(i) When a capacitor of capacitance C farad is charged by connecting it in series with a resistance R ohm and a battery of emf V volts, the charge Q coulomb on the capacitor after a time t seconds is given by

Q =Q₀ (1– e^–t/RC)

where Q₀ = CV which is the final maximum charge (at infinite time) on the capacitor and ‘e’ is the base of natural logarithm. [Q = Q₀ exp(–t/RC)].

The product RC is the time constant of the RC circuit and has dimensions of time. If R is in ohm and C is in farad, RC is in seconds.

(ii) When a capacitor of capacitance C farad having initial charge Q₀ coulomb is discharged through a resistance R ohm, the charge Q coulomb on the capacitor after a time t seconds is given by

Q =Q₀ e^–t/RC

[Or, Q = Q₀ exp(–t/RC)].

The time constant of the RC circuit can be defined using the charging process as well as the discharge process. It is the time required for the charge to build up to [1– (1/e)] times (which is 63.2%) the final maximum charge. It can be defined also as the time required for the charge to decay to (1/e) times (which is 36.8%) the initial charge.

[The growth and decay of charge on a capacitor in a CR circuit is similar to the growth and decay of current through an inductor in an LR circuit. If an inductor of inductance L and a resistor of resistance R are connected in series with a battery, the nature of increase of current I through the circuit is exponential with time and is given by

I = I₀ (1– e^–Rt/L) where I₀ is the final (maximum) current in the circuit.

On disconnecting the battery, the current in the circuit decays exponentially with time as given by the equation

I = I₀ e^–Rt/L

Here L/R is the time constant of the LR circuit].

Questions in this section will be discussed in due course. Meanwhile, find some useful multiple choice questions (with solution) in this section at physicsplus: Multiple Choice Questions on Direct Current Circuits