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 find the current through the conductor at the instant \( t = 2 \) seconds, we can follow these steps:

### Step 1: Understand the relationship between charge and current
The current \( I \) through a conductor is defined as the rate of flow of charge. Mathematically, this can be expressed as:
\[ I = \frac{dQ}{dt} \]
where \( Q \) is the charge.

### Step 2: Differentiate the charge function
We are given the charge \( Q \) as a function of time \( t \):
\[ Q = 2t^2 + 5t \]
To find the current, we need to differentiate \( Q \) with respect to \( t \):
\[ \frac{dQ}{dt} = \frac{d}{dt}(2t^2 + 5t) \]

### Step 3: Apply the differentiation
Using the power rule of differentiation:
- The derivative of \( 2t^2 \) is \( 4t \).
- The derivative of \( 5t \) is \( 5 \).

Thus, we have:
\[ \frac{dQ}{dt} = 4t + 5 \]

### Step 4: Substitute \( t = 2 \) seconds into the current equation
Now, we need to find the current at \( t = 2 \) seconds:
\[ I = 4(2) + 5 \]

### Step 5: Calculate the current
Calculating the above expression:
\[ I = 8 + 5 = 13 \, \text{A} \]

### Conclusion
The current through the conductor at \( t = 2 \) seconds is \( 13 \, \text{A} \).

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