Question 166: Current Electricity

Question

The electric current through a wire varies with time as $$I=I_0+\beta t$$, where $$I_0=20 \mathrm{~A}$$ and $$\beta=3 \mathrm{~A} / \mathrm{s}$$. The amount of electric charge crossed through a section of the wire in $$20 \mathrm{~s}$$ is :

Accepted Answer

To calculate the amount of electric charge $Q$ that crosses through a section of the wire over a period of $20$ seconds, given the current varies with time as $I = I_0 + \beta t$, where $I_0 = 20$ A (initial current) and $\beta = 3$ A/s (rate of change of current with time), we use the concept of integration from calculus because the current is not constant but changes linearly with time.
The electric charge $Q$ is the integral of current $I$ over the time interval from $0$ to $20$ seconds. The formula for $Q$ is given by:
$Q = \int_{0}^{T} I(t) \, dt$
Where $I(t) = I_0 + \beta t$ and $T = 20$ s. Substituting the given values:
$Q = \int_{0}^{20} (20 + 3t) \, dt$
This integral can be solved in two parts:
1. The integral of the constant term $20$:
$\int 20 \, dt = 20t$
2. The integral of the linear term $3t$:
$\int 3t \, dt = \frac{3}{2}t^2$
Thus, combining these and evaluating from $0$ to $20$ seconds:
$Q = \left[20t + \frac{3}{2}t^2\right]_{0}^{20}$
$Q = \left[20(20) + \frac{3}{2}(20)^2\right] - \left[20(0) + \frac{3}{2}(0)^2\right]$
$Q = 400 + 600 = 1000$ C
Therefore, the amount of electric charge that crosses through a section of the wire in $20$ seconds is $1000$ C.

Answer

80 C

Answer

800 C

Answer

1000 C

Answer

1600 C

Current Electricity

The electric current through a wire varies with time as $I=I_0+\beta t, where I_0=20 \mathrm{~A} and \beta=3 \mathrm{~A} / \mathrm{s}. The amount of electric charge crossed through a section of the wire in 20 \mathrm{~s}$ is :

Choose the correct answer: