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

Options

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

Answer

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

### Step 1: Identify the charge function
The charge \( Q \) through a cross-section of the conductor is given by:
\[ Q(t) = 5t^2 - 2t \, \text{coulombs} \]

### Step 2: Calculate the charge at \( t_2 = 4 \, \text{s} \)
Substituting \( t = 4 \) into the charge function:
\[ Q(4) = 5(4^2) - 2(4) \]
Calculating \( 4^2 \):
\[ 4^2 = 16 \]
Now substituting back:
\[ Q(4) = 5(16) - 8 = 80 - 8 = 72 \, \text{coulombs} \]

### Step 3: Calculate the charge at \( t_1 = 2 \, \text{s} \)
Substituting \( t = 2 \) into the charge function:
\[ Q(2) = 5(2^2) - 2(2) \]
Calculating \( 2^2 \):
\[ 2^2 = 4 \]
Now substituting back:
\[ Q(2) = 5(4) - 4 = 20 - 4 = 16 \, \text{coulombs} \]

### Step 4: Calculate the change in charge
The change in charge \( \Delta Q \) from \( t_1 \) to \( t_2 \) is:
\[ \Delta Q = Q(4) - Q(2) = 72 - 16 = 56 \, \text{coulombs} \]

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

### Step 6: Calculate the average current
The average current \( I \) is given by:
\[ I = \frac{\Delta Q}{\Delta t} = \frac{56 \, \text{coulombs}}{2 \, \text{s}} = 28 \, \text{A} \]

### Final Answer
The average current through the conductor in the interval from \( t_1 = 2 \, \text{s} \) to \( t_2 = 4 \, \text{s} \) is:
\[ I = 28 \, \text{A} \]