An electric current in a conductor varies with time according to the expression I(t) = 86 sin (40πt), where I is in amperes and t is in seconds. What is the total charge passing a given point in the conductor from t = 0 to t = 1/60 s?

1.03C

The current (I) flowing through a conductor is related to the charge (Q) passing through the conductor at a given range of time, t, as follows;

I(t) =                 -----------------(i)

Now, to express the charge (Q) in terms of I and t, we need to integrate equation (i) as follows;

dQ = I(t)dt

Q =                  ---------------------------(ii)

Where, according to the question;

I(t) = 86 sin (40πt)

a = final time t = 1/60s

b = initial time t = 0

Substitute these values into equation(ii) as follows;

Q =    

Q = 86 x            ------------------(iii)

Solve the integral as follows;

let u = 40πt           [differentiate both sides]

=> = 40π

=> dt =

=> du = 40π dt

Substitute the values of u = 40πt  and dt = into equation(iii)

Q = 86 x        -------------------(iv)

Q = x

Now, integrate sin u

Q = x [-cos (u)]               ------------------(v)

Substitute the value of u back into equation (v)

Q = x [-cos (40πt)]

Q = - x [cos (40πt)]               ------------------(vi)

Now, insert the values of t = 0 and t = 1/60 into equation (vi) as follows;

Q = - x [  [cos (40π(1/60))]  - [cos (40π(0))]  ]

Q = - x [  [cos ()]  - [cos (0)]  ]

Q = - x [  [cos ()]  - [cos (0)]  ]

Q = - x [  [-0.5]  - [1]  ]

Q = - x [  -1.5 ]    

Q = x [1.5 ]    

Q =

Take π = 3.142

Q =

Q =

Q = 1.03C

Therefore, the total charge passing a given point in the conductor from t = 0 to t = 1/60s is 1.03C