Charge through a cross-section of a conductor is given by Q = `(2t^(2)+5t)C` Find the current through the conductor at the instant t = 2s.

Options

A. 13A
B. `-2A`
C. 2A
D. `-13A`

Answer

To solve the problem, we need to find the current through the conductor at the instant \( t = 2 \) seconds, given the charge \( Q \) as a function of time \( t \).

### Step-by-Step Solution:

1. **Identify the formula for current**:
The current \( I \) through a conductor is defined as the rate of change of charge \( Q \) with respect to time \( t \). Mathematically, this is expressed as:
\[ I = \frac{dQ}{dt} \]

2. **Write the expression for charge**:
The charge \( Q \) is given by the equation:
\[ Q = 2t^2 + 5t \quad \text{(in Coulombs)} \]

3. **Differentiate the charge with respect to time**:
To find the current, we need to differentiate \( Q \) with respect to \( t \):
\[ I = \frac{dQ}{dt} = \frac{d}{dt}(2t^2 + 5t) \]
Using the power rule of differentiation:
\[ \frac{d}{dt}(t^n) = nt^{n-1} \]
We differentiate each term:
- The derivative of \( 2t^2 \) is \( 4t \).
- The derivative of \( 5t \) is \( 5 \).

Therefore, we have:
\[ I = 4t + 5 \]

4. **Substitute \( t = 2 \) seconds into the current expression**:
Now, we need to find the current at \( t = 2 \) seconds:
\[ I(2) = 4(2) + 5 \]
Calculating this gives:
\[ I(2) = 8 + 5 = 13 \, \text{Amperes} \]

5. **Final answer**:
The current through the conductor at \( t = 2 \) seconds is:
\[ I = 13 \, \text{A} \]