Capacitors & Exponential Decay

Unit Structure 1.Electric and magnetic fields 2.Capacitors and exponential decay 3.Nuclear physics 4.Medical imaging 5.Modelling the universe

Capacitors Define Capacitance Capacitance = charge per unit potential difference

Capacitors Define the Farad 1 coulomb per volt ( CV -1 ) Notice that the unit is defined in terms of two other units

Capacitors Define time constant = CR Make sure that C is in Farads and R in Ohms

Capacitors Describe Charge Flow Electrons move anticlockwise Electrons deposited on Y and removed from X An equal number of electrons removed and deposited on the plates.

Capacitors Series or Parallel when S2 closed? Parallel – Positive plate connected to positive plate, negative plate to negative plate.

Capacitors What is the voltage across each capacitor ? Charge on each is the same and Q=CV C 150 x V 150 = C 450 x V 450 C 150 / C 450 = V 450 / V /450 = 1/3 = V 450 / V 150 so 4.5 V across 150 μ F Capacitor Most candidates thought that the ratio of voltages = ratio of capacitors but it is not!

Capacitors Analyse the performance of Capacitors Graphically – What’s special about Exponential Decay The value of I falls by a constant ratio in equal time intervals

Capacitors Analyse the performance of Capacitors Graphically – Sketching an Exponential Decay Work out where it starts Get the shape right Consider if it has to go through any particular point ( often calculated in a previous part of a question)

Capacitors Analyse the performance of Capacitors Graphically – Exponential Decay – finding the time constant Time constant is the time taken for V or I or Q to fall to 1/e of its initial value So 6/e = 6/2.7 = 2.2 V So from graph time constant = 20s

Capacitors Analyse the performance of Capacitors Numerically- The Initial Conditions clue Initially we know that V = 12V and I = 3.24 μA so to find R we can use R = V/I ( Ohms Law) R = 12/3.24 x = 3.7 M Ω Find R ?

Capacitors Analyse the performance of Capacitors Graphically- Voltage vs Charge Gradient = Area under graph = 1/C Energy stored or work done charging the capacitor Be careful to check which way the axes are !

Capacitors Analyse the performance of Capacitors Numerically – Q = CV A very simple equation to use, often at the start of a question, just make sure that you get the units correct.

Capacitors Analyse the performance of Capacitors Numerically – Q = It ( the constant current clue!) A 0.1μ F capacitor is charged at a constant rate with a steady current of 40mA for a time of 60 s.Calculate the final charge. Q = It = 40 x x 60 = 2.4 C

Capacitors Analyse the performance of Capacitors Numerically – x = x o e –t/CR If you have to use ln then ln (x/x o ) = -t/CR

Capacitors Analyse the performance of Capacitors Numerically – The charge and replace question This type of question usually involves a capacitor being charged and then connected into another combination of capacitors Remember Charge is conserved – it stays the same