An electric current in a conductor varies with time according to the expression I(t) = 116 sin (90𝜋t), 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 = 1/180 s?

The total charge passing a given point in the conductor can be found by integrating the current over the given time interval. The charge Q is given by the integral of I(t) from 0 to 1/180 s.

The integral of I(t) from 0 to 1/180 s is:

∫ from 0 to 1/180 of 116 sin(90πt) dt

This is a standard integral, the integral of sin(at) from 0 to b is -cos(at)/a evaluated from 0 to b.

So, ∫ from 0 to 1/180 of 116 sin(90πt) dt = [-116 cos(90πt) / (90π)] from 0 to 1/180

Evaluating this at 1/180 gives -116 cos(90π * 1/180) / (90π) = -116 cos(π/2) / (90π)

And evaluating this at 0 gives -116 cos(90π * 0) / (90π) = -116 cos(0) / (90π)

The total charge Q is then the difference between these two values:

Q = [-116 cos(π/2) / (90π)] - [-116 cos(0) / (90π)]

cos(π/2) = 0 and cos(0) = 1, so

Q = [-116 * 0 / (90π)] - [-116 * 1 / (90π)] = 116 / (90π) = 0.41 Coulombs

So, the total charge passing a given point in the conductor from t = 0 to t = 1/180 s is approximately 0.41 Coulombs.