Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Capacitance Year 13 What is a capacitor? • A device which stores energy in an electric field • Two conductors insulated from one another Capacitors Charging a capacitor • When connected to a voltage source, electrons flow onto one plate and off the other, producing a charge difference • The higher the charge already on the plate, the harder it is to add even more Charging a capacitor • Constant voltage power supply • As the capacitor becomes charged, the current drops V V A B +q -q Electron flow +Q -Q Fully charged. No current Capacitance • We define the capacitance of a capacitor as the charge stored per unit potential difference: Q C , or Q CV V • The unit of capacitance is the Farad (F) • Most devices have values ranging from pF to mF • Real devices also have a defined maximum voltage, above which they break down Air tank analogy Air tank analogy Discharging a capacitor • When a charged capacitor is connected to a resistance, electrons flow to rebalance the charge distribution • As the charge difference reduces, so the potential difference drops V A Capacitance of a parallel plate capacitor [not examinable] – e0 is the permitivity of free space – K is the relative permitivity of any dielectric between the plates Capacitance modified by dielectric [not examinable] • Applied field polarises the dielectric. • This decreases the effective field between the plates. • Easier to add more charge - this increases the capacitance. • Note that the dielectric should be an insulator! Capacitors store energy • How do we know that the charged capacitor stores energy? • How did the capacitor gain its energy store? V Energy stored by a capacitor • It gets progressively harder to add more charge to an Potential difference already charged capacitor V • Energy stored = work done charging • Work done adding DQ is VDQ • Work done charging capacitor to V is area under graph 1 1 E QV CV 2 2 2 DQ Charge Q Energy supplied by battery • V=E/Q, so Energy from battery=QV • Energy stored by capacitor =1/2 QV • So where did the other 1/2 go? • Dissipated by parasitic resistance – This is always true, regardless of the value fo the resistance Example • A 10 mF capacitor is charged to 20 V. How much energy is stored? • Energy stored = 1/2 CV2 = 2000 mJ • Calculate the energy stored at 10 V (i.e. at half the voltage): • 500 mJ, one quarter of the previous value (EaV2) Charging and discharging capacitors Capacitor Discharge • Switch to A: capacitor charges • Switch to B: capacitor discharges • At a time t, charge is Q, pd is V and current is I: I=V/R around circuit I=-DQ/Dt in a short time Dt V=Q/C for capacitor So DQ=-(Q/CR) Dt DQ 1 Dt Q RC dQ 1 or Q dt RC C A B I R DQ 1 Dt Q RC Q t 1 1 Q Q dQ RC 0 dt 0 1 t ln Q t 0 RC t ln Q ln Q0 RC Q Q0 Q Q0 e t RC Exponential decay Time constant, RC 1 When t=RC, Q Q0 e i.e. Q=0.368Q0 RC is known as the Time constant Q=CV It is the time taken for the charge (and V=IR therefore also the voltage and current) to drop to 1/e of its initial value • In another period RC, the charge will have dropped to 1/e2 of its initial value • etc... • cf radioactive decay constant • • • • Discharging Capacitor Capacitor Discharge Summary: Charging capacitors [not examinable] Capacitor Discharge Other exponentials in nature • Air leaking from a bike tyre • Radiation absorption with distance • Decay of earthquake wave amplitude with distance • Sand through an egg timer • Population trends • Moore’s law for semiconductors • ... Capacitors in parallel • • • • Both have the same voltage V QT=Q1+Q2 But Q=CV, so: CTV=C1V+C2V, or • CT=C1+C2 • …it’s just like increasing the size of the plates… Capacitors in series • Both have the same charging current, so each has the same displaced charge Q • V=V1+V2 • V=Q/C, so • Q/CT=Q/C1+Q/C2 • So 1 1 1 CT C1 C2