Question

Suppose the current in a conductor decreases exponentially with time according to the equation I(t) = I₀ e**(-t / T) , where I₀ is the initial current (at. t = 0 ) and T is a constant. having dimensions of time. Consider a fixed observation point within the conductor. (b) How much charge passes this point between t=0 and t=10 \tau ?

Answer 1

1. The quantity of charge that passes the point between t = 0 and t = τ is 0.6321I₀τ.
2. The quantity of charge that passes the point between t = 0 and t = 10τ is 0.99995I₀τ.
3. The quantity of charge that passes the point between t = 0 and t = ∞ is I₀τ.

How to determine the charge?

In an electric field, Euler's number suggest that current starts with a maximum value and decreases to 36.9% of the preceding numerical value in every time interval, τ.

Also, the current in this conductor decreases exponentially with time and as such, it would be used to derive an equation for charge through integration:

I = dQ/dt

dQ = Idt

By integrating, we have this general integral:

Q = ∫dQ = ∫Idt

At an end-time, T, the integral becomes:

$Q = \int\limits^T_0 {I_0e^{\frac{-t}{\tau} } } \, dt$

From time, t = 0 to time, t = T, we would integrate to have:

$Q = \int\limits^T_0 {-I_0\tau)e^{\frac{-t}{\tau} } } \, (\frac{dt}{\tau} )$

At t = 0, we set Q = 0 and obtain:

$Q = {-I_0\tau(e^{\frac{-t}{\tau} } - e^0)} \, (\frac{dt}{\tau} )\\\\Q = I_0\tau(1-e^{\frac{-t}{\tau} })$

Between t = 0 and t = τ, the charge that passes through this point is given by:

Q = I₀τ(1 - e⁻¹)

Q = 0.6321I₀τ.

Between t = 0 and t = 10τ, the charge that passes through this point is given by:

Q = I₀τ(1 - e⁻¹⁰)

Q = 0.99995I₀τ.

Between t = 0 and t = ∞, the charge that passes through this point is given by:

Q = I₀τ(1 - e⁻∞)

Q = I₀τ.

In conclusion, we can reasonably infer and logically deduce that the amount of current decreases sharply with respect to time and this makes the total charge it transports is finite.

Read more on current and charge here: brainly.com/question/15693063

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Complete Question:

Suppose the current in a conductor decreases exponentially with time according to the equation I(t) = I₀e^(-t / T) , where I₀ is the initial current (at. t = 0 ) and T is a constant. having dimensions of time. Consider a fixed observation point within the conductor.

(a) How much charge passes this point between t = 0 and t = τ?

(b) How much charge passes this point between t = 0 and t = 10τ?

(c) How much charge passes this point between t = 0 and t = ∞?