Understanding Engineering Mathematics

4.In a circuit with an inductor of inductanceLand capacitor of capacitanceC, the voltage

across the inductance is given by

V=L

dI

dt

whereIis the current through the inductance, and the current through the capacitor is

given by

I=C

dV

dt

whereVis the potential across the capacitor.

(i) The time dependence of the current in a seriesRLcircuit with a constant voltage

sourceEcan be accurately modelled by the expression

I=

E

R

[

1 −exp

(

−

Rt

L

)]

What is the voltage across the inductor at timet?

(ii) AnRCcircuit with a constant capacitor and constant voltage source includes a

resistor whose resistance varies slowly according to a linear lawR=R 0 ( 1 +αt).

The voltage across the capacitor in such circumstances can be modelled by the

relation

V−V 0 =(E−V 0 )[1−( 1 +αt)−^1 /αCR^0 ]

whereV 0 is the voltage across the capacitor att=0. Determine the expression

for the current through the capacitor at timet.

5.The potential in the junction region of a semiconductorpnjunction is given by

V=

A

1 +exp(−kx)

whereAandkare material-dependent constants. The force on an electron with charge

ein such a potential region is given by

F=e

dV

dx

Obtain an expression for this force in terms of hyperbolic functions (138

➤

).

Answers to reinforcement exercises

8.3.1 Geometrical interpretation of differentiation

A.(i) 2 (ii)−1 (iii) 2 (iv)− 3
B.x=1and− 2