2.1: Capacitors

Explore

1. Choose a color for the wire connected to the negative side of the battery and call that \(0V\). Use that same color for all three circuits.
2. Color code the remaining wires different colors for different voltage values.
3. Complete the table to compare the three circuits.

Critical Thinking & Concept Invention

As a group, discuss and complete the following statements:

• Given enough time, \(\Delta V_{battery}\) and \(\Delta V_{Capacitor}\) __________________.
• As the voltage across a capacitor increases, the amount of charge on each plate __________________.

\(Q \propto \Delta V_C\)

OR

\(Q \propto \frac{1}{\Delta V_C}\)

• The total net charge on both plates of a capacitor is __________________.
• This means the amount of positive charge on one plate __________________.

Summary

Capacitors are devices that can accumulate charge. The amount of charge a capacitor can hold is based on the electric potential difference (voltage) applied to the two plates of the capacitor.

\(Q_C = C\Delta V_C\)

The capacitance (\(C\)) of a capacitor represents the amount of charge per volt that a capacitor can accumulate and is a property of the geometric shape and materials used in the capacitor.

The unit for capacitance is Farads (F), which are equivalent to Coulombs per Volt:

\(F=\frac{C}{V}\)

We will explore those details later in the course. If we rearrange this equation to solve for the voltage across the capacitor, we can use capacitor voltage in Kirchhoff Voltage Rule equations.

\(\Delta V_C=\frac{Q}{C}\)

The sign of this voltage difference follows the same rule as batteries: the positive side is a higher voltage than the negative side.

Explore

1. Draw a Kirchhoff path in the direction that current will flow, based on the battery terminals.
2. Construct a Voltage Equation using Kirchhoff’s Voltage Rule for the circuit and put a (+) or (−) sign over each \(\Delta V\) term. You can use:
   • \(\Delta V_S\) for the voltage source
   • \(\Delta V_R\) for the voltage difference across the resistor
   • \(\Delta V_C\) for the voltage difference between the two plates of the capacitor

Critical Thinking & Concept Invention

1. Do the charges that go through the resistor (as current) make a complete loop around the circuit? If not, where do they end up? Explain your thinking.
2. Use Ohm’s Relation and the equation for capacitor voltage to determine the potential difference across the resistor (\(\Delta V_R\)) and the current through the resistor (\(I\)) in each of the three circuit snapshots. Be careful with signs (+/−).
3. Summarize how voltages, currents, and charges relate in RC circuits:

If \(Q_C=0\), then:

\(\Delta V_R=\) ____________ and \(I_R=\) ____________

If \(\Delta V_C=-\Delta V_S\), then:

\(\Delta V_R=\) ____________ and \(I_R=\) ____________

This value for \(I_R\) makes sense when a capacitor is “fully charged” because __________________.

Summary

Capacitors are a break in the circuit where charge can accumulate. Current can flow through wires to either plate of a capacitor, but current does not actually flow through a capacitor.

The current going through wires connected to a capacitor represents the rate that charges are added to or removed from the capacitor. Negative charges flow toward or away from the plates of a capacitor until there is no longer a voltage drop between the battery terminal and the capacitor plate.

Explore

1. On the left model (a), highlight each wire a different color, including the plate of the capacitor that is connected to that wire.
2. On the right model (b), highlight each wire using the same two colors connected to the positive and negative battery terminals as in model (a).
3. Circle the wires (if any) that have a net charge of zero:

\(\Sigma Q = 0\)

4. Which wires (if any) do not have a net charge of zero?

\(\Sigma Q \ne 0\)

Critical Thinking & Concept Invention

1. How does the amount of charge on each capacitor in series compare? Why is this necessary?
2. How does the charge on the series capacitors compare to the charge of the single capacitor in model (b)?
3. Compare the original series circuit (a) to the equivalent circuit (b):
   • Construct a Voltage Equation using Kirchhoff’s Voltage Rule for each.
   • Plug in the appropriate \(\frac{Q}{C}\) for each voltage term (note the subscripts and assume capacitances may be different).
   • Solve the system of two equations to generate an equation for \(C_S\) in terms of \(C_1\), \(C_2\), and \(C_3\).
   • You may review equivalent resistance relationships for guidance.

Explore

1. On the top model (a), highlight each wire a different color, including the plate of the capacitor connected to that wire. Remember that color does not change at junctions.
2. On the bottom model (b), highlight each wire using the same two colors connected to the positive and negative battery terminals as in model (a).
3. Draw Kirchhoff paths in the direction current would flow from the battery. Draw enough paths so there is a path for every capacitor.
4. Circle the wires (if any) that have a net charge of zero:

\(\Sigma Q = 0\)

5. Which wires (if any) do not have a net charge of zero?

\(\Sigma Q \ne 0\)

Critical Thinking & Concept Invention

1. Which is the same for each capacitor in model (a): the charge (\(Q\)) or the voltage difference (\(\Delta V\))?
2. Compare the original parallel circuit (a) to the equivalent circuit (b):
   • Construct a set of Voltage Equations using Kirchhoff’s Voltage Rule for each capacitor.
   • Plug in the appropriate \(\frac{Q}{C}\) for each voltage term (note the subscripts and assume capacitances may be different).
   • Solve the system of equations to generate an equation for \(C_P\) in terms of \(C_1\), \(C_2\), and \(C_3\).
   • You may review equivalent resistance relationships for guidance.

Summary

The patterns for equivalent capacitance are the inverse of the patterns for equivalent resistance.

Similar to equivalent resistance:

• You can only simplify a set of capacitors in series to the equivalent series capacitance if __________________________________.
• You can only simplify a set of capacitors in parallel to the equivalent parallel capacitance if __________________________________.