Electric Current as a function of time

[jmuduke]

Homework Statement

8. An electric current in a conductor varies with time according to the expression
I(t) = 100 sin (120*pi*t), 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 = 1/240 s?

Homework Equations

The Attempt at a Solution

I have attempted to substitute the values of t into the equation and use the difference, but I do not feel that was the correct way. Next, I attempted to perform a definite integral, but I get 0 for both numbers.

 

[tiny-tim]

welcome to pf!

hi jmuduke! welcome to pf!

I(t) = 100 sin (120*pi*t), 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 = 1/240 s?
…
I attempted to perform a definite integral, but I get 0 for both numbers.

yes, I = dQ/dt, so Q = ∫ I dt

it's only over 90°, so you shouldn't get 0 for both limits

show us what you did ​

 

[jmuduke]

Thanks for the reply Tim!

I calculated the integral and got -(5 cos(120*pi*t))/6*pi

Originally, my calculator was set in radians, so that could have been why I got 0 for both limits. I changed it to degrees and got -0.265 for both limits then, but that result in the definite integral being 0, correct?

 

[tiny-tim]

hi jmuduke!

(just got up :zzz:)

I calculated the integral and got -(5 cos(120*pi*t))/6*pi

… got -0.265 for both limits then, but that result in the definite integral being 0, correct?

yes, cos(0) = 1, so that's correct for the t = 0 limit

but for t = 1/240, cos(120πt) = cos(π/2) = cos90° = 0 ​

 

[asteriatic]

I would approach this problem by first understanding the physical concept behind the given expression. Electric current is the rate of flow of electric charge, and it is typically measured in amperes (A). In this case, the current varies with time according to a sinusoidal function, where the amplitude is 100 A and the frequency is 120*pi Hz.

To find the total charge passing through a given point in the conductor, we can use the definition of electric current as the rate of change of charge with respect to time: I = dQ/dt. Rearranging this equation, we get dQ = I*dt.

Using this relationship, we can integrate the given expression for I(t) from t = 0 to t = 1/240 s to find the total charge passing through the point in that time interval. This gives us:

Q = ∫I(t)dt = ∫100 sin(120*pi*t) dt = (-100/120*pi) cos(120*pi*t) from t = 0 to t = 1/240 s

Plugging in the values, we get Q = (-100/120*pi) cos(1/2) - (-100/120*pi) cos(0) = 0.0441 C

Therefore, the total charge passing through the given point in the conductor from t = 0 to t = 1/240 s is 0.0441 coulombs. This approach takes into account the changing current with time and gives a more accurate answer compared to simply using the difference in values.