PHYC10004 Chap.5 Current, Resistance and Kirchhoff's Laws
Current, Resistance and Kirchhoff's Laws
Week 4 moves from static charge to moving charge. Current is the rate at which charge crosses a section of a conductor, i = dq/dt = enAv_d, and resistance R = V/i measures how strongly scattering in the material resists that drift.
A device obeys Ohm's law only if R stays constant across voltages, and the power dissipated in any element is P = iV.
The heart of the chapter is Kirchhoff's method for networks: the junction rule, which is conservation of charge, and the loop rule, which is conservation of energy. With a current labelled in every branch and a consistent sign convention, any DC network reduces to simultaneous equations.
The chapter adds the practical side from the laboratory programme, including internal resistance, meter loading, the RC time constant and the Wheatstone bridge, and it explains why bulbs in different arrangements glow with different brightness. None of the network rules is printed on the formula sheet, yet every recent paper carries a circuits question.
What this chapter covers
- 01
Current and Drift Velocity
- 02
Resistance and Resistivity
- 03
Ohm's Law and Ohmic Devices
- 04
Electrical Power
- 05
Kirchhoff's Junction Rule
- 06
Kirchhoff's Loop Rule
- 07
Internal Resistance
- 08
RC Time Constant
- 09
Wheatstone Bridge Balance
Worked example · free
Terminal voltage and bulb power from a battery with internal resistance
- 1Parallel pair: 1/R = 1/6.0 + 1/6.0, so R = 3.0 Ω.
- 1Loop rule round the whole circuit: 12 − 0.50i − 3.0i = 0, so i = 3.4 A.
- 1Terminal voltage: V = ε − ir = 12 − 0.50 × 3.43 = 10.3 V.
- 1Each resistor has the full terminal voltage across it: P = V²/R = 10.3²/6.0 = 17.6 W.
- 1Energy check: 2 × 17.6 W in the load plus i²r = 5.9 W inside the battery equals εi = 41 W.
Key terms
- Electric Current
- Electric current is the rate at which charge passes through a cross-section of a conductor, i = dq/dt, measured in amperes.
- Drift Velocity
- Drift velocity is the small average velocity that free charge carriers acquire along the field in a conductor, superimposed on their fast random motion.
- Resistivity
- Resistivity is the property of a material that sets its resistance for a given shape; for metals it rises with temperature as ρ = ρ₀(1 + αΔT).
- Ohmic Device
- An ohmic device has a resistance that stays the same whatever voltage is applied or current flows; a carbon resistor is ohmic and a diode is not.
- Internal Resistance
- Internal resistance is the resistance inside a real source, which makes its terminal voltage fall below the emf as the current rises.
- Kirchhoff's Loop Rule
- Kirchhoff's loop rule states that the potential changes around any closed loop of a circuit add to zero, a statement of energy conservation.
- Time Constant
- The time constant of an RC circuit is τ = RC, the time for a discharging capacitor's charge to fall to about 37% of its starting value.
Current, Resistance and Kirchhoff's Laws FAQ
Why do bulbs in parallel glow brighter than the same bulbs in series?
Brightness follows the power in each bulb, P = V²/R. In parallel each bulb receives the full battery voltage; in series the voltage is shared between them, so each bulb's power falls sharply even though the current is the same through both.
What does a negative current mean in a Kirchhoff solution?
It means the current flows opposite to the arrow you drew. Keep the original equations, report the magnitude with the corrected direction, and if that branch contains a battery check whether the battery is now being charged rather than delivering energy.
How does a capacitor behave in a DC circuit?
At the instant a switch closes, an uncharged capacitor acts like a plain wire. As it charges the current through it falls exponentially with time constant RC, and once fully charged it carries no current, behaving like a break in the circuit.
Can one voltage and current reading show a device is ohmic?
No. Ohm's law requires the resistance to stay constant across many voltages or currents. A single reading only gives V/i at that point, so several measurements are needed before you can call the device ohmic.
Exam move
Practise Kirchhoff's method on networks of rising difficulty: one battery and four resistors, two batteries sharing a middle branch, and a circuit containing a capacitor that you analyse just after switching and long afterwards. Always label every branch current, write the junction equations, then walk each loop with the sign rule, and check your answer by finding one potential difference along two different paths.
Alongside the algebra, rehearse short explanations: bulb brightness, why a switch between equal potentials changes nothing, internal resistance from two load readings, and the balance condition of a Wheatstone bridge.
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