Section 10: Capacitors
Uses of capacitors
-A camera flash as the camera battery charges the capacitor over a few seconds, and then the entire charge of the capacitor is dumped into the flash almost instantly. -This allows the flash to be very bright for a very short time. -Ultracapacitors (really big capacitors) can be used in back-up power supplies to provide reliable power for short periods of time. -To smooth out variations in d.c. voltage supplies a capacitor can be used to absorbs the peaks and fills in the troughs.
Capacitor
-A capacitor is an electrical component that can store electrical charge. -They are made up of two electrical conducting plates separated by an electrical insulator (a dielectric).
Permittivity
-Altering the properties of a capacitor changes how much charge it can store at a given (its capacitance). -One of the things you can change is the dielectric material separating the two conducting plates. -This changes the capacitance because different materials have different relative permittivities. -Permittivity is a measure of how difficult it is to generate an electric field in a medium. -The higher the permittivity of a material, the more charge is needed to generate an electric field of a given size. Relative permittivity is the ratio of the permittivity of a material to the permittivity of free space: Er= E1/E0 (Er= relative permittivity of material 1) (E1= permittivity of material 1 in Fm-1) (E0= permittivity of free space= 8.85 x10^-12 Fm^-1) Relative permittivity is sometimes also called the dielectric constant.
Why can't a capacitor be used as a battery?
-Capacitors can only store relatively small amounts of charge. -Capacitors are usually used to provide power for a short amount of ti me as it is tricky to prolong the discharge time and the voltage through the circuit decreases as the capacitor discharges.
Log-linear graphs
-Log-linear graphs are plots where one of the axes is logarithmic. -Log-linear and log-log plots are useful as they can often be used to produce a graph which is a straight line when linear axes would give a curve. -They're a good way of graphically displaying how charge, potential difference and current vary over time for a discharging capacitor.
Polar molecules
-Permittivity can be explained by the motion (or action) of the molecules inside a dielectric. -Imagine that a dielectric is made up of lots of polar molecules. -When no charge is being stored by a capacitor, no electric field is being generated. -The molecules are aligned randomly. -When charge is applied to the plates of a capacitor an electric field is generated between them. -The negative ends of the molecules are attracted to the positively charged plate, and vice versa. -This causes the molecules to rotate and align themselves anti-parallel to the electric field generated between the plates. -The molecules each have their own electric field, which in this alignment now opposes the applied electric field of the capacitor. -The larger the permittivity , the larger this opposing field is. -This reduces the overall electric field between the parallel plates, which reduces the potential difference needed to transfer a given charge to the capacitor (so the capacitance increases Q= CV).
The capacitance of a capacitor
-The capacitance of a capacitor is the charge that the capacitor can store per unit potential difference across it. -The voltage rating of a capacitor is the maximum potential difference that can be safely put across it. -A capacitor will only charge up to the voltage of the power source it is connected to.
Capacitance
-The capacitance of an object is the amount of charge it is able to store per unit potential difference (p.d) across it. -Capacitance is measured in farads. 1 farad= 1 coulomb per volt (CV^-1) C= Q/V (C= capacitance in F) (Q= charge in C) (V= potential difference in V)
Why are capacitors useful yet harmful?
-They can store charge until it's needed, and then discharge all of their charge in a fraction of a second, whereas a battery might take several minutes. -For this reason, charged capacitors can be very dangerous. -Capacitors in electronics in your home could contain enough charge to kill you.
Discharging through a fixed resistor
-To discharge a capacitor, take out the battery and reconnect the circuit. -When a charged capacitor is connected across a resistor, the p.d. drives a current through the circuit. -This current flows in the opposite direction from the charging current. -The capacitor is fully discharged when the p.d. across the plates and the current in the circuit are both zero.
Energy stored by capacitors
-When a capacitor charges, one plate becomes negatively charged while the other becomes positively charged. -Like charges repel, so when each plate of a capacitor becomes charged, the charges on that plate are being forced together 'against their will'. -This requires energy, which is supplied by the power source and stored as electric potential energy for as long as the charges are held. -When the charges are released, the electric potential energy is released. E= 1/2 QV E= 1/2 CV^2 E= (1/2) (Q^2/C) (E= energy stored in J) (Q= charge on capacitor in C) (V= potential difference across capacitor in V) (C= capacitance in F)
Charging
-When a capacitor is connected to a d.c. power supply a current flows in the circuit until the capacitor is fully charged, then stops. -The electrons flow from the negative terminal of the supply onto the plate connected to it, so a negative charge builds up on that plate. -At the same time, electrons flow from the other plate to the positive terminal of the supply, making that plate positive. -These electrons are repelled by the negative charge on the negative plate and attracted to the positive terminal of the supply. -The same number of electrons are repelled from the positive plate as are built up on the negative plate. -This means an equal but opposite charge builds up on each plate, causing the potential difference between the plates. -No charge can flow directly between the plates because they're separated by an insulator (dielectric). -Initially the current through the circuit is high. -But, as charge builds up on the plates, electrostatic repulsion makes it harder and harder for more electrons to be deposited. -When the p.d. across the capacitor is equal to the p.d. across the supply, the current falls to zero, -The capacitor is fully charged.
What happens when a capacitor is connected to a direct current power source?
-When a capacitor is connected to a direct current (d.c) power source, charge builds up on its plates. -One becomes positively charged and one plate becomes negatively charged. -The plates are separated by an electrical insulator so no charge can move between them. -This means that a potential difference builds up between the plates of the capacitor.
Investigating capacitance
-You can investigate how capacitance changes by setting up two parallel plates separated by a dielectric and connecting the plates to a capacitance meter. -You can then after how much the two plates overlap to change the effective area of the capacitor, or use different materials as the dielectric to vary the relative permittivity. -Stacking multiple layer of the same material allows you to test how plate separation affects the capacitance. -If you know the capacitance of a capacitor, the area of its plates and the distance between plates, you could also calculate the relative permittivity of a range of different dielectrics.
Calculating capacitance
C= (A)(E0)(Er)/(d) (C= capacitance in F) (A= effective area of a plate, in m3) (E0= permittivity of free space, in Fm-1) (Er= relative permittivity) (d= distance between the capacitor plates in m)
Charging through a fixed resistor
If you charge a capacitor through a fixed resistor, the resistance of the resistor will affect the time taken to charge the capacitor. The charge on a charging capacitor after a given time t is given by: Q= Q0(1 - e(to the power of (- t/RC) ) (Q= charge of the capacitor at time t, in C) (Q0= charge of the capacitor in C when fully charged) (R= resistance of fixed resistor in ohms) (t= time since charging began in s) (C= capacitance of capacitor in f) The formula for calculating the voltage across a charging capacitor is of the same form, but the formula for the charging current is different (it decreases exponentially). V= V0(1 - e(to the power of (- t/RC) ) I= I0e(to the power of (- t/RC) ) (I0 is the intial current and I is the current at time t, in A)
Time constant
The time t = RC is known as the time constant and is the time taken for the charge on a discharging capacitor (Q) to fall to about 37% of Q0.
Charging and discharging times
The time taken for a capacitor to charge and discharge depends on two factors: -The capacitance of the capacitor (C). This affects the amount of charge that can be transferred at a given voltage. -The resistance of the circuit (R). This affects the current in the circuit.
Time to halve
The time to halve is the time taken for the charge, current or potential difference of a discharging capacitor to derease to half of the intital value. T1/2 = 0.69RC (T1/2= time to halve in s) (C= capacitance of the capacitor in F) (R= resistance of fixed resistor in ohms)