Charge through a cross section of a conductor is give by `Q = 5t^(2) - 2t` coulomb .Find the average current through the conduction the interval `t_(1) = 2 s ` to `t_(2) = 4s` .

Options

A. 14A
B. 28A
C. 56A
D. 7A

Explanation

To find the average current through a conductor over the interval from \( t_1 = 2 \, \text{s} \) to \( t_2 = 4 \, \text{s} \), we can follow these steps:

### Step 1: Write down the expression for charge
The charge \( Q \) through a cross-section of the conductor is given by the equation:
\[ Q(t) = 5t^2 - 2t \]

### Step 2: Calculate the charge at \( t = 4 \, \text{s} \)
To find the charge at \( t = 4 \, \text{s} \), substitute \( t = 4 \) into the equation:
\[ Q(4) = 5(4^2) - 2(4) \]
Calculating this gives:
\[ Q(4) = 5(16) - 8 = 80 - 8 = 72 \, \text{C} \]

### Step 3: Calculate the charge at \( t = 2 \, \text{s} \)
Now, substitute \( t = 2 \) into the equation to find the charge at that time:
\[ Q(2) = 5(2^2) - 2(2) \]
Calculating this gives:
\[ Q(2) = 5(4) - 4 = 20 - 4 = 16 \, \text{C} \]

### Step 4: Find the change in charge (\( \Delta Q \))
The change in charge over the interval from \( t_1 \) to \( t_2 \) is given by:
\[ \Delta Q = Q(4) - Q(2) = 72 \, \text{C} - 16 \, \text{C} = 56 \, \text{C} \]

### Step 5: Calculate the time interval (\( \Delta t \))
The time interval from \( t_1 \) to \( t_2 \) is:
\[ \Delta t = t_2 - t_1 = 4 \, \text{s} - 2 \, \text{s} = 2 \, \text{s} \]

### Step 6: Calculate the average current (\( I_{\text{avg}} \))
The average current is defined as the change in charge divided by the change in time:
\[ I_{\text{avg}} = \frac{\Delta Q}{\Delta t} = \frac{56 \, \text{C}}{2 \, \text{s}} = 28 \, \text{A} \]

### Final Answer
The average current through the conductor from \( t = 2 \, \text{s} \) to \( t = 4 \, \text{s} \) is:
\[ I_{\text{avg}} = 28 \, \text{A} \]

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