Capacitance

A capacitor is an electrical component that stores energy in an electric field. It typically consists of two conductive plates separated by an insulating material called a dielectric. When connected to a power source like a battery, positive charge builds up on one plate and an equal amount of negative charge builds up on the other. Capacitance (C) is a measure of a capacitor's ability to store charge. It is defined as the amount of charge (Q) stored on each plate for every one volt of potential difference (V) across the plates. The relationship is given by the formula Q = CV. The unit of capacitance is the farad (F). One farad is a very large unit, so capacitance is often measured in microfarads (1 μF = 10⁻⁶ F), nanofarads (1 nF = 10⁻⁹ F), or picofarads (1 pF = 10⁻¹² F).

To charge a capacitor, work must be done to move electrons from the positive plate to the negative plate against the growing potential difference. This work is stored as electric potential energy in the electric field between the plates. We can find the energy stored by considering the graph of potential difference (V) against charge (Q). As charge is added, the potential difference increases proportionally (V = Q/C), so the graph is a straight line through the origin. The total energy stored (W) is equal to the area under this graph. The area of the triangle is ½ × base × height, which gives W = ½ QV. By substituting Q = CV or V = Q/C, we can derive two other useful forms of the energy equation: W = ½ CV² and W = ½ Q²/C. You should be able to use and derive all three forms.

When capacitors are connected in parallel, the 'top' plates of all capacitors are connected together, and the 'bottom' plates are all connected together. This means the potential difference (V) across each capacitor is the same as the supply voltage. The total charge stored (Q_total) is the sum of the charges stored on each individual capacitor (Q_total = Q₁ + Q₂ + ...). Since Q = CV, we can write C_total * V = C₁V + C₂V + ... . Because V is the same for all terms, we can divide it out, leaving the simple rule for combining capacitors in parallel: C_total = C₁ + C₂ + ... . The total capacitance is simply the sum of the individual capacitances. This means adding a capacitor in parallel always increases the total capacitance of the circuit.

When capacitors are connected in series, they are placed one after another in a single line. In this arrangement, the key principles are: 1) The total potential difference from the supply (V_total) is shared between the capacitors (V_total = V₁ + V₂ + ...). 2) The charge (Q) stored on each capacitor is the same. This is because the charging process effectively moves charge from the outer plate of the first capacitor to the outer plate of the last one, inducing equal and opposite charges on the inner plates. Since V = Q/C, we can substitute this into the voltage equation: Q/C_total = Q/C₁ + Q/C₂ + ... . The charge Q is the same for all terms, so we can cancel it, giving the rule for capacitors in series: 1/C_total = 1/C₁ + 1/C₂ + ... . Remember to take the reciprocal of your final answer to find C_total. Adding a capacitor in series always decreases the total capacitance.

When a charged capacitor is connected across a resistor, it discharges. Electrons flow from the negative plate, through the resistor, to the positive plate until the plates are neutral. This flow of charge is a current. At the start (t=0), the potential difference (V) and charge (Q) are at their maximum, so the initial current (I = V/R) is also at its maximum. As the capacitor discharges, Q decreases, which causes V to decrease (since V=Q/C). As V decreases, the current I also decreases (since I=V/R). The result is that the charge, voltage, and current all fall over time with a distinctive curve called an exponential decay. The rate of this decay is determined by the 'time constant' of the circuit, given the symbol τ (tau). The time constant is the product of the resistance and capacitance: τ = CR. It represents the time taken for the charge, voltage, or current to fall to 1/e (approximately 37%) of its initial value. A large time constant (from a large R or C) means a slow discharge.

The exponential decay of a discharging capacitor can be described precisely with a single mathematical formula. For any quantity 'x' (which can be charge Q, potential difference V, or current I) that is decaying, its value at time 't' is given by: x = x₀e^(-t/RC). In this equation, x₀ is the initial value of the quantity at t=0, 'e' is the base of the natural logarithm (approximately 2.718), and RC is the time constant (τ) of the circuit. This equation allows you to calculate the charge, voltage, or current at any moment during the discharge. For example, the potential difference is V = V₀e^(-t/RC). You can also rearrange this equation to find the time it takes for a quantity to fall to a certain level. This requires using natural logarithms (ln). Taking the natural log of both sides of V/V₀ = e^(-t/RC) gives ln(V/V₀) = -t/RC, which can be rearranged to find t: t = -RC ln(V/V₀).