Understand the basic laws of current, voltage, and resistance so that you can analyze complex circuits.
Basic Knowledge
Ohm's law: V = IR
・V: voltage [V], I: current [A], R: resistance [Ω]
Connections of resistors:
・Series connection: R = R₁ + R₂ (the equivalent resistance is the sum of the resistances), and the current through each resistor is the same
・Parallel connection: 1/R = 1/R₁ + 1/R₂ (the equivalent resistance is smaller than each resistance), and the voltage across each resistor is the same
Power and electrical energy:
・Power: P = VI = I²R = V²/R [W (watt)]
・Electrical energy: W = Pt [J] or [Wh]
Kirchhoff's laws:
・First law (current law): The sum of currents entering a junction = the sum of currents leaving it
・Second law (voltage law): The sum of voltage drops in a closed loop = the sum of electromotive forces
Ohm's law (Ohm's law)（V = IR. The basic relationship among voltage, current, and resistance）
Electrical resistance (R)（the difficulty with which current flows. Unit: Ω (ohm)）
Electric power (P)（electrical energy per unit time. P = VI [W]）
Kirchhoff's laws（the current law and voltage law used to solve complex circuits）
Internal resistance（the resistance of the battery itself. For electromotive force E and internal resistance r, terminal voltage V = E − Ir）
Joule heat（heat generated by current. Q = I²Rt [J]）
Further Exploration (Background and Significance)
When a battery's internal resistance is considered, if the resistance of the external circuit is R, the current is I = E/(R + r), and the terminal voltage is V = E − Ir = ER/(R + r). As the current increases, the voltage drop across the internal resistance increases, and the terminal voltage decreases. Battery “degradation” is an increase in internal resistance.
Resistivity ρ (rho): R = ρL/S (L: length, S: cross-sectional area). Long, thin conductors have greater resistance, while thick, short conductors have less resistance. For metals, resistivity increases as temperature rises (the positive temperature coefficient of metals).
In Kirchhoff's second law, equations are set up while moving around a loop in one direction, treating “the battery's electromotive force as + and voltage drops as −.” Even complex circuits with multiple electromotive forces can be solved systematically.
- Ohm's law: V = IR
- Series: R = R₁ + R₂, and the current is the same
- Parallel: 1/R = 1/R₁ + 1/R₂, and the voltage is the same
- Power: P = VI = I²R = V²/R
- Joule heat: Q = I²Rt
- Kirchhoff's first law: the sum of currents at a junction = 0
- Internal resistance: terminal voltage V = E − Ir
Notes (Common Confusions)
① The equivalent resistance increases in series and decreases in parallel. This is opposite to capacitors. ② Power P = VI, I²R, and V²/R are all equivalent. Choose the most convenient form. ③ Signs are easy to get wrong in Kirchhoff's laws. Setting a direction for the current and then forming equations gives consistency. ④ Terminal voltage is smaller than the electromotive force because of the voltage drop across the internal resistance.
Practice
- Resistors of 3 Ω and 6 Ω are connected in parallel to a 12 V power source. Find the total current in the circuit.
- When a current of 2 A flows through a 10 Ω resistor, find the power consumed.
- A battery with electromotive force 6 V and internal resistance 1 Ω is connected to an external resistance of 5 Ω. Find the terminal voltage.