Charge through a conductor is given as a function of time t as q=4t^2 +4t +4` coulomb. At 2s what is the current flowing ?

Options

A. 12A
B. 8A
C. 20A
D. 28 A

Answer

To find the current flowing through the conductor at \( t = 2 \) seconds, we need to follow these steps:

### Step 1: Understand the relationship between charge and current
The current \( I \) flowing through a conductor is defined as the rate of change of charge \( q \) with respect to time \( t \):
\[ I = \frac{dq}{dt} \]

### Step 2: Differentiate the charge function
Given the charge function:
\[ q(t) = 4t^2 + 4t + 4 \]
we need to differentiate this function with respect to \( t \) to find the current.

### Step 3: Perform the differentiation
Using the power rule of differentiation:
\[ \frac{d}{dt}(t^n) = n \cdot t^{n-1} \]
we differentiate \( q(t) \):
\[ \frac{dq}{dt} = \frac{d}{dt}(4t^2) + \frac{d}{dt}(4t) + \frac{d}{dt}(4) \]
Calculating each term:
- The derivative of \( 4t^2 \) is \( 8t \).
- The derivative of \( 4t \) is \( 4 \).
- The derivative of a constant \( 4 \) is \( 0 \).

Putting it all together:
\[ \frac{dq}{dt} = 8t + 4 \]

### Step 4: Substitute \( t = 2 \) seconds into the current equation
Now we can find the current at \( t = 2 \) seconds:
\[ I = 8(2) + 4 \]
Calculating this gives:
\[ I = 16 + 4 = 20 \, \text{A} \]

### Final Answer
The current flowing at \( t = 2 \) seconds is \( 20 \, \text{A} \).

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