The Sign of the Charge Carriers

• Positive charges move in the same direction of the current.
• Negative charges move in the opposite direction of the current.
• Current always flows from high potential to low potential, independent of the sign of the charge carriers.
• Postive charges move from high potential to low potential, while negative charges climb from low potential to high potential.
• Electric field (not shown) always points in the same direction as the current.

Electric Potential in a Circuit

• Battery pushes up the electric potential to $\mathcal{E}$, the emf of the battery (usually just called the voltage of the battery, in the absence of internal resistance).
• As current flows across a resistor, the electric potential drops by $IR$.

Internal Resistance

• The shaded area represents a non-deal (real) battery with internal resistance $r$.
• $r$ causes the potential to drop by $Ir$, so the apparent voltage $V_{apparent}$ of the battery is less than the true emf $\mathcal{E}$.
• The voltmeters above display only the magnitude of the potential difference but not the signs.
• $V_{apparent}$ is close to $\mathcal{E}$ if $r$ is small compared to $R$. Adjust $r$ and $R$ in the simulation to see.
• Can you explain why increasing $R$ while keeping $r$ the same increases $V_{apparent}$?

Combination of Resistors

• Resistors in series have increased resistance.
• Resistors in parallel have reduced resistance.
• For 3 resistors:
• Series: $R_{123} = R_1 + R_2 + R_3$.
• Parallel: $R_{123} = (R_1^{-1} + R_2^{-1} + R_3^{-1})^{-1}$.

Circuit Step by Step

1. Resistors in series have increased resistance.
2. Resistors in parallel have reduced resistance.
3. For 3 resistors:
• Series: $R_{123} = R_1 + R_2 + R_3$.
• Parallel: $R_{123} = (R_1^{-1} + R_2^{-1} + R_3^{-1})^{-1}$.

Circuits with Adjustable Resistance

• Click on a resistor to see details and adjust resistance.
• Click on the battery to reveal the total current and resistance of the circuit.
• Observe:
  • Resistors in series always have the same current.
  • Branches in parallel always have the same total voltage.

Parallel Plate Capacitor

• Not draw to scale. Plate separation exaggerated for clarity.
• $1pF=10^{-12}F$

Capacitor at Constant Charge / Voltage

• Both capacitors were charged to $2V$. One of them is then disconnected from the battery.
• If the separation is changed, the capacitor without the battery will have constant charge $q$ because there is nowhere for the charge to go. Its voltage $V_C$ will change.
• The one connected to the battery has changing charge $q$ but a fixed voltage $V_C$. A battery can be thought of as a resevoir of charges, so when the capacitance goes up, charges flow from the battery to the capacitor and vice versa. This is an idealization. In reality, it takes time to equalize the capacitor voltage with the emf of the battery. See the $RC$ simulation below for a more realistic depiction.
• The intensity of the colors on the capacitor plates reflect the amount of charge: red for positive, blue for negative.
• Not draw to scale. Plate separation exaggerated for clarity.
• $1pF=10^{-12}F$

Combination of Capacitors

• Capacitors in series have reduced capacitance.
• Capacitors in parallel have increased capacitance.
• For 3 capacitors:
• Series: $C_{123} = (C_1^{-1} + C_2^{-1} + C_3^{-1})^{-1}$.
• Parallel: $C_{123} = C_1 + C_2 + C_3$.

RC Circuit

• Play to charge capacitor. To discharge, set the battery voltage to $0V$.
• Observe how rapidly the capacitor charges when you change the resistance and capacitance.
• The intensity of the colors on the capacitor plates reflect the amount of charge: red for positive, blue for negative.
• $1\mu F=10^{-6}F$

Equations:

• $\mathcal{E} - IR - \frac{q}{C} = 0$ (Kirchhoff's loop rule)

Virtual lab:

• Drag the coordinate tool (the square box) to measure the half time $T_{1/2}$ when the charge is half of the maximum value.
• Use $T_{1/2} = \tau \ln 2$ to calculate the time constant $\tau$ and compare it with the theory $\tau = RC$.

RL Circuit

• Play to "charge" inductor. To "discharge", set the battery voltage to $0V$.
• Observe how rapidly the current rises when you change the resistance and inductance.

Equations:

• $\mathcal{E} - IR - L\frac{dI}{dt} = 0$ (Kirchhoff's loop rule)

Virtual lab:

• Drag the coordinate tool (the square box) to measure the half time $T_{1/2}$ when the charge is half of the maximum value.
• Use $T_{1/2} = \tau \ln 2$ to calculate the time constant $\tau$ and compare it with the theory $\tau = \frac{L}{R}$.