Question

In a region electric field is given by `vec E = 4 x hat i` N/C. The potential difference between points A(x = 1 m) and B(x = 3 m), (V_A − V_B) is ______.

विकल्प

• −16 V

• 16 V

• −8 V

• 8 V

Answer 1

In a region electric field is given by `vec E = 4 x hat i` N/C. The potential difference between points A(x = 1 m) and B(x = 3 m), (V_A − V_B) is 16 V.

Explanation:

The potential difference (V_A − V_B) between two points in an electric field `vec E` is related to the work done per unit charge by the field as it moves from A to B. The general formula for potential difference is:

V_B − V_A = `-int_(x_A)^(x_B) vec E * d vec x`

To find V_A − V_B, we rearrange the expression:

V_A − V_B = `-int_(x_A)^(x_B) vec E * d vec x`

Given the electric field `vec E = 4 x hat i` N/C and the points A at x = 1 m and B at x = 3 m. 

Substitute these into the integral:

V_A − V_B = `int_1^3 4x dx`

Integrate the function 4x w.r.t. x:

∫4x dx = `4[x^2/2]`

= 2x²

Now, apply the limits from 1 to 3:

V_A − V_B = `2[x^2]_1^3`

= 2(3² − 1²)

= 2(9 − 1)

= 2(8)

= 16 V

∴ The potential difference (V_A − V_B) is 16 V.