Question

the electric potential at a certain point is given by v = - 7.5x² + 3x, where v is in volts and x is in meters. what is the electric field at that point?
○ e = 0
○ e = (- 2.5x³ + 1.5x²)î
○ e = (2.5x³ - 1.5x²)î
○ e = (- 15x + 3)î
○ e = (15x - 3)î

Explanation:

Step1: Recall the relationship between electric - field and potential

The electric - field $\vec{E}$ and electric potential $V$ are related by $\vec{E}=-
abla V$. In one - dimension (since $V$ is a function of $x$ only), $E_x =-\frac{dV}{dx}$.

Step2: Differentiate the given potential function

Given $V=-7.5x^{2}+3x$. Using the power - rule of differentiation $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $\frac{dV}{dx}=\frac{d}{dx}(-7.5x^{2}+3x)=-15x + 3$.

Step3: Find the electric - field

Since $\vec{E}=-E_x\hat{i}$, and $E_x =-\frac{dV}{dx}$, then $\vec{E}=(-15x + 3)\hat{i}$.

Answer:

$\vec{E}=(-15x + 3)\hat{i}$