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Capacitance
Capacitance
Capacitors are electrical devices used to store energy in electronic circuits, commonly for a backup release of energy if the power fails They are in the form of two conductive metal plates connected to a voltage supply (parallel plate capacitor) o There is commonly a dielectric in between the plates, this is to ensure charge does not freely flow between the plates Capacitors are marked with a value of their capacitance. This is defined as:
The charge stored per unit potential difference (between the
plates)
The greater the capacitance , the greater the energy stored in the capacitor Polar Molecule in an Electric Field A dielectric is made up of many polar molecules o These are molecules that have a 'positive' and 'negative' end (poles) When no charge is applied to the capacitor: o There is no electric field between the parallel plates and the molecules are aligned in random directions When there is a charge applied : o One of the parallel plates becomes positively charged and the other negatively charged hence an electric field is generated between the plates (from positive to negative) o The negative ends of the polar molecules are attracted to the positive plate and vice versa o This means all the molecules rotate and align themselves parallel to the electric field
Parallel Plate Capacitor Permittivity is the measure of how easy it is to generate an electric field in a certain material
The relativity permittivity εr is sometimes known as the dielectric
constant For a given material, it is defined as: The ratio of the permittivity of a material to the permittivity of free space This can be expressed as: o εr = ε / ε 0 o εr = relative permittivity o ε = permittivity of a material (F m -1) o ε 0 = permittivity of free space (F m-1) Dielectric Action in a Capacitor When the polar molecules in a dielectric align with the applied electric field from the plates, they each produce their own electric field o This electric field opposes the electric field from the plates The larger the opposing electric field from the polar molecules in the dielectric, the larger the permittivity o In other words, the permittivity is how well the polar molecules in a dielectric align with an applied electric field The opposing electric field reduces the overall electric field, which decreases the potential difference between the plates o Therefore, the capacitance of the plates increases The capacitance of a capacitor can also be written in terms of the relative permittivity: o C = Aε 0 εr / d o C = capacitance (F)
The electrical (potential) energy stored in the capacitor can be determined from the area under the potential-charge graph Therefore, the work done, or energy stored , by a capacitor is defined by the equation: o E = 1/2 QV
o E = work done or energy stored (J)
o Q = charge (C)
o V = potential difference (V)
Substituting the charge with the capacitance equation Q = CV , the
energy stored can also be defined as: o E = 1/2 CV^2
By substituting the potential V, the energy stored can also be
defined in terms of just the charge, Q and the capacitance, C:
o E = Q^2 / 2C
Capacitor Charge & Discharge
Charge & Discharge Graphs
Charging Capacitors are charged by a power supply (eg. a battery) When charging, the electrons are pulled from the plate connected to the positive terminal of the power supply o Hence the plate nearest the positive terminal is positively charged They travel around the circuit and are pushed onto the plate connected to the negative terminal o Hence the plate nearest the negative terminal is negatively charged
As the negative charge builds up, fewer electrons are pushed onto the plate due to electrostatic repulsion from the electrons already on the plate When no more electrons can be pushed onto the negative plate, the charging stops At the start of charging, the current is large and gradually falls to zero as the electrons stop flowing through the circuit o The current decreases exponentially o This means the rate at which the current decreases is proportional to the amount of current it has left Since an equal but opposite charge builds up on each plate, the potential difference between the plates slowly increases until it is the same as that of the power supply Similarly, the charge of the plates slowly increases until it is at its maximum charge defined by the capacitance of the capacitor The key features of the charging graphs are: o The shapes of the p.d. and charge against time graphs are identical o The current against time graph is an exponential decay curve o The initial value of the current starts on the y axis and decreases exponentially o The initial value of the p.d and charge starts at 0 up to a maximum value Discharging Capacitors are discharged through a resistor with no power supply present The electrons now flow back from the negative plate to the positive plate until there are equal numbers on each plate and no potential difference between them
o I = current (A)
o Δ Q = change in charge (C)
o Δ t = change in time (s)
This means that the area under a current-time graph for a charging (or discharging) capacitor is the charge stored for a certain time interval In the discharging graph, this is the discharging current at that time In the charging graph, this is the charging current at that time To calculate the gradient of a curve, draw a tangent at that point and calculate the gradient of that tangent Time Constant The time constant of a capacitor discharging through a resistor is a measure of how long it takes for the capacitor to discharge The definition of the time constant is: o The time taken for the charge, current or voltage of a discharging capacitor to decrease to 37% of its original value o Alternatively, for a charging capacitor: The time taken for the charge or voltage of a charging capacitor to rise to 63% of its maximum value o 37% is 0.37 or 1 / e (where e is the exponential function)
multiplied by the original value ( I 0 , Q 0 or V 0 )
o This is represented by the Greek letter tau, , and measured in units of seconds (s) It is a useful way of comparing the rate of change of similar quantities e.g. charge, current or p.d. o It is defined by the equation:
o T = RC
o Where: T = time constant (s)
R = resistance of the resistor (Ω)
C = capacitance of the capacitor (F)
Charge & Discharge Equations The time constant is used in the exponential decay equations for the current, charge or potential difference (p.d) for a capacitor discharging through a resistor o These can be used to determine the amount of current, charge or p.d left after a certain amount of time for a discharging capacitor This exponential decay means that no matter how much charge is initially on the plates, the amount of time it takes for that charge to halve is the same The exponential decay of current on a discharging capacitor is defined by the equation: o I = I 0 e -t/RC Where:
o I = current (A)
o I 0 = initial current before discharge (A)
o e = the exponential function
o t = time (s)
o RC = resistance (Ω) × capacitance (F) = the time constant τ
(s) This equation shows that the smaller the time constant τ, the quicker the exponential decay of the current when discharging Also, how big the initial current is affects the rate of discharge
o If I 0 is large, the capacitor will take longer to discharge
The current at any time is directly proportional to the p.d across the capacitor and the charge across the parallel plates Therefore, this equation also describes the charge on the capacitor after a certain amount of time: Q = Q 0 e -t/RC
The charging equation for the current I is the same as its
discharging equation, since the current still decreases exponentially
