Question

The charge at a point in space is q (t) = 0.5 cos (6t) C. What is the current at t = −2 s? At an arbitrary time t?

Answer 1

Solution:

Given: A charge at a point in space whose magnitude is changing with time; that is amount of charge q at that point is a function of time t and it is given by:

q(t) = 0.5*cos(6t) C

We know that the current i is can be obtained by taking the derivative of the function q(t) with respect to time t;

i = di/dt A                           (unit: ampere A = coulomb/sec = C/s)

i = d[0.5*cos(6t)]/dt A

derivative of cos(6t) with time is d(cos(6t)/dt = -sin(6t)d(6t)/dt = -6*sin(6t)

thus current i is,

i = 0.5*[-6*sin(6t)] A

i = -3*sin(6t) A

Above expression gives us the value of current at any time t.

Since this current i also depend on time t, must denote the current i as i(t). Thus above expression can be written as,

i(t) = -3*sin(6t) A             -------------------------------------------------------------------------------------(1)

Note that the angle 6t in sin(6t) as whole must have dimension of angle in radians. 6 must be the constant (generally ω = angular speed) whose dimensions are rad/sec.

The current at time t = -2 s is,

i(-2) = -3*sin(6*-2) A

Putting calculator in radian mode, we get

i(-2) = -1.61 A.

Thus the current at time t = -2s is i = -1.61 A.

The current at an arbitrary time t is i(t) = -3*sin(6t) A.