The quantity Q of charge in coulombs [tex]$C$[/tex] that has passed through a point in a wire up to time [tex]$T$[/tex] (measured in seconds) is given by [tex]$Q(t)=t^3-3 t^2+4 t+3$[/tex]. Find the current when [tex]$T=4 s$[/tex].

Find the derivative of the charge function Q ( t ) to get the current function I ( t ) .
Differentiate Q ( t ) = t 3 − 3 t 2 + 4 t + 3 to find I ( t ) = 3 t 2 − 6 t + 4 .
Substitute t = 4 into the current function: I ( 4 ) = 3 ( 4 ) 2 − 6 ( 4 ) + 4 .
Calculate the current at t = 4 : I ( 4 ) = 28 . The current when t = 4 is 28 ​ .

Explanation

Problem Analysis We are given the charge Q ( t ) = t 3 − 3 t 2 + 4 t + 3 and asked to find the current at time t = 4 seconds. The current is the rate of change of charge with respect to time, so we need to find the derivative of Q ( t ) with respect to t .

Finding the Current To find the current I ( t ) , we differentiate Q ( t ) with respect to t :


I ( t ) = d t d Q ​ = d t d ​ ( t 3 − 3 t 2 + 4 t + 3 )

Differentiating Q(t) Using the power rule for differentiation, we get:

I ( t ) = 3 t 2 − 6 t + 4

Evaluating I(4) Now, we need to find the current at t = 4 seconds. We substitute t = 4 into the expression for I ( t ) :

I ( 4 ) = 3 ( 4 ) 2 − 6 ( 4 ) + 4

Calculating the Current Calculating the value, we have:

I ( 4 ) = 3 ( 16 ) − 24 + 4 = 48 − 24 + 4 = 24 + 4 = 28

Final Answer Therefore, the current at t = 4 seconds is 28 Coulombs per second (Amperes).

Examples
Understanding current and charge flow is crucial in electrical engineering. For example, when designing a circuit, engineers need to know how the charge changes over time to ensure that the components are working correctly and that the circuit is safe. The relationship between charge and current is also important in medical devices, such as pacemakers, where precise control of electrical signals is essential for proper function.

Answered by GinnyAnswer | 2025-07-05