Question

The current in a conductor varies with time t as I = 2t+3tsqare, where I is in ampere t in second. What is d electric charge flowing through a cross-section of d conductor during t=2s to t=3s?

Answer 1

The current in a conductor varies with time t as I = 2t + 3t² , where I is in Ampere and t is in second.

we know, current is rate of flowing of charge per unit time.

i.e., instantaneous current, I = dQ/dt

or, $\int{I}\,dt=Q$

then, charge flowing through a cross section during t = 2s to 3s , Q = $\int\limits^3_2{(2t+3t)^2}\,dt$

= $2\int\limits^3_2{t}\,dt+3\int\limits^3_2{t^2}\,dt$

= $[t^2]^3_2+[t^3]^3_2$

= [3² - 2²] + [3³ - 2³]

= (9 - 4) + (27 - 8)

= 5 + 19

= 24 C

hence, charge flowing through conductor is 24C.

Answer 2

Explanation:

The current in a conductor varies with time t as I = 2t + 3t² , where I is in Ampere and t is in second.

we know, current is rate of flowing of charge per unit time.

i.e., instantaneous current, I = dQ/dt

or, \int{I}\,dt=Q∫Idt=Q

then, charge flowing through a cross section during t = 2s to 3s , Q = \int\limits^3_2{(2t+3t)^2}\,dt

2

∫

3

(2t+3t)

2

dt

= 2\int\limits^3_2{t}\,dt+3\int\limits^3_2{t^2}\,dt2

2

∫

3

tdt+3

2

∫

3

t

2

dt

= [t^2]^3_2+[t^3]^3_2[t

2

]

2

3

+[t

3

]

2

3

= [3² - 2²] + [3³ - 2³]

= (9 - 4) + (27 - 8)

= 5 + 19

= 24 C

hence, charge flowing through conductor is 24C