
Electricity
"Proof" Exercises Question 1
A battery has an emf,
, and internal resistance, r. The battery is connected to a variable resistor, R, as shown below. 

a) 
As the variable resistor is varied, the power dissipated in it will also vary. Show that the power, P, dissipated in the variable resistor is given by 

b) 
If
= 9V and r = 3W, what is the maximum value of the power dissipated in the variable resistor, R? 

You might know a mathematical method for finding the maximum value of a function. If not, you will have to make a guess as to what value of R will result in the maximum power being supplied to it.
If R >> r then the power dissipated in the variable resistor is very low (try, say, R = 300 ...); similarly, if R << r, P is very low (try, for example, R = 0·03 ...). So, how about R = ??... 

Question 2
An electron of charge, e, and mass, m, enters the uniform electric field between two parallel charged plates, as shown in the diagram below. 

The p.d. across the plates is V and the distance between them is d. The initial velocity of the electron is of magnitude, u and is parallel to the plates. The point of entry into the field is midway between the two plates. The electron hits the lower plate after having moved a distance x in the field (x measured parallel to the plates). Show that 


