Current in a conductor is expressed as I = 8t^3 + 3t^2 + 1 , where current I is measured in amperes and time t is measured in seconds. What is the charge that flows through a cross-section of the conductor between time t = 1s to t = 2s?

Step-by-step solution

The current in the conductor is given by I = dq dt = 8t^3 + 3t^2 + 1 The charge q that flows through a cross-section is the integral of current with respect to time. dq = I dt = (8t^3 + 3t^2 + 1) dt Integrating both sides from t = 1 s to t = 2 s: q = ₁² (8t^3 + 3t^2 + 1) dt q = [ 8t^4 4 + 3t^3 3 + t ]₁² q = [ 2t^4 + t^3 + t ]₁² q = (2(2)^4 + (2)^3 + 2) - (2(1)^4 + (1)^3 + 1) q = (2(16) + 8 + 2) - (2 + 1 + 1) q = (32 + 8 + 2) - 4 q = 42 - 4 = 38 C