1.1. Pacemakers are devices which are used to deliver short pulses of charge to a patient who is suffering from a cardiac arrest. 

Figure 1 shows a circuit  that is used in such a pacemaker. 

Figure 1

The capacitor is initially uncharged when at t = 0 s, then the two-way switch connects it to a power supply of 4.0 kV. 

Sketch the graph on the axes provided in Figure 2 that shows how the voltage across the capacitor varies over 10 seconds. 

Figure 2

1.2. Show that the time taken for the charge stored in the capacitor to reach 5.0 mC is approximately 0.3 s.  

1.3. Pacemaker manufacturers need efficient ways of measuring how quickly the charge and energy are released by capacitors. 

One approach uses the ‘exponential factor’ σ, which is a ratio of the energy remaining in a capacitor E to its initial energy stored ${E}_{0}$ . For a given time interval, a smaller ratio indicates a quicker greater discharge of energy. 

Show that the exponential factor σ is given by:

 σ = ${e}^{-\frac{2t}{RC}}$