capacitors 1

Derivation for I = Q/RC x -t

I = V/R

I = Q/C/R = Q/CR

Q = Q/CR x -t

Negative used to show the decreasing charge.

Derivation for the time taken for something to become 1/e of it’s original value.

I = Io e^-t/CR

Io/e = Io e^-t/CR

1/e = e^-t/CR

e^-1 = e^-t/CR

-1 = -t/CR

t = CR

Show that the half life t/2 is related to the time constant CR by t/2 = CRln(2)

I = Io e-0.5t/CR

Io/2 = Io e-0.5t/CR

½ = e^-0.5t/CR

Ln(1/2) =-0.5t/CR

-Ln(1/2) = 0.5t/CR

Ln(2) = 0.5t/CR

CRLn(2) = 0.5t

capacitance

a measurement for the charge stored per volt.

time constant

T=RC time constant is the time for a value to drop by 37% of its value.

constant ratio property of exponentials

y=y0​ekt is the equation used to find the change in value with respect to time. Exponential graphs always drop by the same ratio in the sm=ame time e.g 60% in 40 seconds.

methods of determining the time constant from graphs

1. to find where the value is 37% lower than the previous value as that is time for T.

2. logarithm method, plot a graph of ln(y) against t, where the gradient = -1/T

3. constant ratio method, chose an equal time interval e.g 5 seconds then use y(t)/y(t+Δt)​ and adjust Δt until the ratio is close to 0.37 (e^-1)

combining capacitors

when capacitors are combined in parallel they are added normally as their capacitances add up, resulting in a total capacitance that is greater than any individual capacitor this is because the P.D is the same in each parallel circuit (Kirchoffs’s 2nd law) and since C = Q/V and the charge is constant then

Itotal​=V(R​+R/​)⇒Rt​1​=R1​+R1​.

for series circuits, capacitance is found by using 1/C1 + 1/C2 = 1/Ct as the voltage is split across them (Kirchhoff’s 2nd law) as V = I/R I is constant in a series loop but

Vtotal​=I(R1​+R2​)⇒Req​=R1​+R2​

energy on a capacitor

energy is found by using a graph since energy is = QV then we can plot a graph of Q against V where the gradient is C. the area will be a triangle and this area is energy W. half of the energy is lost when charging a capacitor due to the work done to push the electrons onto the negative plate.