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Flashcards · AP® Physics C: Electricity and Magnetism · Unit 11

Unit 11: Electric Circuits flashcards

61 cards on topics 11.1 to 11.8. Pick a topic or study the whole unit.

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11.1 Electric Current

What is electric current?
The rate at which charge crosses a cross-section: I = dq/dt, in amperes (1 A = 1 C/s). Charge passed is q = ∫I dt, the area under the I–t graph.
How is current related to drift velocity?
I = nqv_dA, where n is carriers per m³, q the carrier charge, v_d the drift speed and A the cross-section. The same current in a wire of three times the area gives one third the drift speed.
What is current density?
J = I/A for uniform flow, in A/m², and as a vector J = nqv_d, pointing the way positive charge flows. For electrons it points opposite to their drift.
How is the field in a conductor related to current density?
E = ρJ, where ρ is resistivity in Ω·m, not charge density.
How do you find current when the current density varies across a wire?
I = ∫J·dA. For a round wire, integrate over rings: I = ∫₀ᴿ J(r) 2πr dr.
Is current a vector?
No. It has a direction along the wire but is a scalar: 2 A and 3 A into a junction give 5 A out.
What is conventional current?
The direction positive charge would move. In metal wires, electrons actually move the opposite way.
Are charges at rest when there is no current?
No. Free electrons move randomly at about 10⁶ m/s. Zero current means zero net flow; a field adds only a tiny drift, often below 1 mm/s.

11.2 Simple Circuits

When does an element in a circuit carry a steady current?
Only when it is in a closed loop that contains a source and has a potential difference across it.
What is the difference between an open circuit and a short circuit?
Open: a break stops the flow, so I = 0 in that path. Short: a path with no change in potential; elements across it carry no current.
What happens to a lit bulb if a wire joins its two ends?
It goes out. The wire shorts it, so there is no potential difference across the bulb.
How do you tell a battery symbol from a capacitor symbol?
Battery: two unequal parallel lines, the long line being +. Capacitor: two equal parallel lines.
What does a circuit schematic show?
What is connected to what, not where things physically sit. Elements can belong to several closed loops, so one change can affect several parts.
Does current get used up around a loop?
No. In a single loop the current is the same at every point. Elements transfer energy, not charge.
Which way does conventional current flow outside a battery?
Out of the + terminal, round the circuit, and into the − terminal. Electron flow is the opposite.

11.3 Resistance, Resistivity, and Ohm’s Law

What is resistance?
R = ΔV/I, in ohms (1 Ω = 1 V/A), measuring how strongly an object opposes the movement of charge. The definition holds for any element at any moment.
What is Ohm's law?
The claim that R is constant for all currents, so ΔV = IR with fixed R. R = ΔV/I alone is just the definition of resistance.
How does resistance depend on a wire's dimensions?
R = ρℓ/A, from E = ρJ, J = I/A and ΔV = Eℓ. Use A = πr² = πd²/4; using the diameter as the radius makes A four times too large.
What is resistivity?
ρ, in Ω·m, a property of the material set by its atomic structure, independent of the sample's shape. In metals it usually rises with temperature because carriers scatter more off vibrating ions.
What happens to resistance when a wire is stretched to n times its length?
Volume is fixed, so A → A/n, and R → n²R.
How do you find resistance when resistivity varies along a wire?
Treat thin slices in series: R = (1/A)∫₀ˡ ρ(x) dx.
How do you get resistance from an I–ΔV graph?
With I on the vertical axis, the slope of a straight line is 1/R. For a non-ohmic element such as a lamp, use ΔV/I at the point, not the tangent slope.
What happens to resistance when a uniform wire is cut in half?
Each half has R/2, since R ∝ ℓ.

11.4 Electric Power

What is electrical power?
P = IΔV for any element, from dU = ΔV dq. 1 W = 1 J/s = 1 V·A. Sources deliver power and loads receive it; total delivered equals total received.
What is the power dissipated in a resistor?
P = I²R = ΔV²/R, converted to thermal energy. For a fixed resistor, doubling ΔV or I quadruples P.
Does more resistance mean more power?
It depends on what is fixed. Same current (series): P ∝ R. Same potential difference (parallel): P ∝ 1/R.
How do you compare bulb brightness?
Brightness increases with power, so compare powers, not currents or resistances alone.
What is the power input to a motor?
IΔV = P_mech + P_thermal. I²R is only the thermal part. Lifting at constant speed needs P_mech = mgv.
How do you find energy transferred when power varies?
E = ∫P dt, the area under the P–t graph. Do not average I or ΔV before squaring.
Is a kilowatt-hour a unit of power?
No. It is energy: 1 kWh = 3.6 × 10⁶ J.
How much energy does a 60 W device transfer in 2.0 minutes?
E = Pt = 60 × 120 = 7.2 × 10³ J.

11.5 Compound Direct Current Circuits

How do you tell whether resistors are in series or in parallel?
Series: all the charge through one must go through the other, with no other path, so they carry the same current. Parallel: they share the same two end junctions, so they have the same potential difference. Check the junctions, not the layout.
What are the equivalent resistances for series and parallel resistors?
Series: R_eq = R₁ + R₂ + …. Parallel: 1/R_eq = 1/R₁ + 1/R₂ + …, then invert. For two in parallel, R_eq = R₁R₂/(R₁ + R₂); n identical resistors in parallel give R/n.
What happens to total current when a resistor is added in parallel?
The extra path lowers R_eq, so with an ideal battery the total current rises.
How do you solve a compound resistor network?
Reduce it from the inside out to one equivalent resistance, find the total current, then work back out to find the current and potential difference for each resistor.
What is a real battery's terminal voltage?
ΔV = ℰ − Ir, where r is the internal resistance. In a single loop, I = ℰ/(R + r). The emf equals the terminal voltage only when no current flows.
How should ammeters and voltmeters be connected, and what are their ideal resistances?
Ammeter in series, ideal resistance zero. Voltmeter in parallel, ideal resistance infinite, so no current passes through it and the circuit is unchanged.
What happens if an ammeter is connected across an element?
It creates a short circuit, because its near-zero resistance gives the current an easy path round the element.
Does a battery supply a constant current?
No. An ideal battery fixes the emf ℰ, and the current depends on the resistance connected.

11.6 Kirchhoff’s Loop Rule

What does Kirchhoff's loop rule state?
The potential differences around any closed loop add to zero: ΣΔV = 0. It is a statement of energy conservation, since a charge returns to the same potential.
What sign do you give a resistor in a loop equation?
−IR when you cross it in the direction of the assumed current, +IR when you cross it against the current.
What sign do you give a battery in a loop equation?
+ℰ when crossing from the − terminal to the + terminal, −ℰ when crossing from + to −. An ideal wire contributes 0.
What does a negative current from a loop equation mean?
The current flows in the opposite direction to the one assumed, with the same size. It is not an error; keep your original arrows.
What is the terminal voltage of a battery being charged?
ΔV_terminal = ℰ + Ir. When it delivers current, ΔV_terminal = ℰ − Ir.
How do you draw a potential-against-position graph for a loop?
A rise of ℰ across a source crossed − to +, a straight-line fall of IR across each uniform resistor in the current direction, flat sections along ideal wires, ending at the starting level.
Can you choose any point to be V = 0 in a circuit?
Yes. Only potential differences are physical, so any point can be the reference.

11.7 Kirchhoff’s Junction Rule

What does Kirchhoff's junction rule state?
ΣI_in = ΣI_out at any junction, because charge cannot build up there. It is conservation of charge, and it also applies to any closed region of a circuit.
How many junction equations are independent?
A circuit with N junctions gives N − 1 independent junction equations. Write loop equations for the remaining unknowns.
How many unknown currents does a circuit have?
One per branch, where a branch runs between two junctions and carries one current throughout. B branches need B equations in total.
What is the method for solving a multi-loop circuit?
Label one current per branch with an assumed direction; write N − 1 junction equations; add loop equations for the rest; solve; interpret negative signs; check with a closed region or an energy balance.
Does the junction rule hold while a capacitor is charging?
Yes, at every instant. Charge collects on the capacitor plates, not at junctions.
Does current split equally at a junction?
Only between identical branches. In general the split depends on each branch's resistance.
Currents of 0.8 A enter a junction; 0.3 A and 0.2 A leave along two other wires. What is the fourth current?
0.8 − 0.3 − 0.2 = 0.3 A, leaving the junction.

11.8 Resistor-Capacitor (RC) Circuits

How do capacitors combine in series and in parallel?
Parallel (same ΔV): C_eq = C₁ + C₂ + …. Series (same Q): 1/C_eq = 1/C₁ + 1/C₂ + …, less than the smallest capacitance. This is the reverse of resistors.
What differential equation describes a charging RC circuit?
The loop rule gives R dq/dt + q/C = ℰ, with I = dq/dt. Its solution is q = Cℰ(1 − e^(−t/RC)) and I = (ℰ/R)e^(−t/RC).
How does a capacitor discharge through a resistor?
R dq/dt + q/C = 0 with I = −dq/dt gives q = Q₀e^(−t/RC). The capacitor voltage and the current decay by the same factor.
What is the time constant of an RC circuit?
τ = RC, in seconds (Ω·F = s). After one τ a charging capacitor reaches 63% of its final charge; a discharging one keeps 37%. Half takes about 0.69τ.
How does an uncharged capacitor behave just after the switch closes?
Like a wire: ΔV_C = 0 and the current is greatest, set by the resistors. An already charged capacitor acts like a battery of its present voltage.
How does a capacitor behave long after the switch closes?
Like a break: no current in its branch, and ΔV_C at its steady-state value. Use this after about 5τ.
How is energy shared when a capacitor charges through a resistor from a battery?
The battery supplies Cℰ²; half (½Cℰ²) is stored in the capacitor and half is dissipated in the resistor.
How can you find RC from discharge data with a straight-line graph?
Plot ln ΔV_C against t: ln ΔV_C = ln ΔV₀ − t/(RC), so the slope is −1/(RC).

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