Question

What is the magnitude of the electric field in a region where the potential is given by the expression V = ax2 + b where a = −500 V/m2 and b = 135 V? (Substitute numerical values, and express your answer in terms of x. Assume SI units, but do not include units in your answer.)

Answer 1

Answer:

The equation of the electric field (E) is given by;

E = -1000x

Explanation:

The electric field (E) is the gradient of the potential difference (V) in the direction of x between two points. i.e

E = $\frac{dV}{dx}$           ----------------------(i)

From the question;

V = ax² + b            ---------------(ii)

Find the gradient of equation (ii) by taking derivative of both sides with respect to x as follows;

$\frac{dV}{dx}$ = $\frac{d(ax^{2} + b)}{dx}$

$\frac{dV}{dx}$ = 2ax               ----------------------(iii)

Substitute $\frac{dV}{dx}$ = 2ax into equation (i) as follows;

E = 2ax                --------------------------(iv)

Now, substitute the value of a = -500 into equation (iv) as follows;

E = 2(-500)x

E = -1000x

Therefore, the equation of the electric field (E) is given by;

E = -1000x

Answer 2

Answer:

E = 1000 x

Explanation:

The electric potential and the electric field are related by the formula

        dV = - E . dx

Bold represents vectors.

The point represents the scalar product, in this case we calculate the electric field in the x-axis and the potential is also in this axis so the scalar product is reduced to the algebraic product

        E = dV /dx

Let's make the derivative

        E = - 2ax

Let's replace the values

        E = -2 (-500) x

        E = 1000 x