© All Rights reserved @ LearnWithDash
Step-by-Step Explanation
Step 1: Recall the Definition of Electric Field in Terms of Potential
The electric field $\vec{E}$ is related to the electric potential $V$ by the relation:
$$ \vec{E} = -\nabla V $$
This means the electric field is the negative gradient (or spatial rate of change) of the electric potential.
Step 2: Identify the Given Information
We are told that the electric potential is constant ($5\,\text{V}$) throughout a certain region of space with volume $0.2\,\text{m}^3$. A constant potential implies that $V(x,y,z) = 5\,\text{V}$ does not change with position.
Step 3: Understand the Implication of Constant Potential
Since $V$ does not vary in space, its gradient (i.e., its rate of change in any direction) is zero:
$$ \nabla V = 0 $$
Step 4: Conclude the Magnitude of the Electric Field
Because the electric field is $\vec{E} = -\nabla V$, if $\nabla V = 0$, then
$$ \vec{E} = 0. $$
Hence, the magnitude of $\vec{E}$ is zero in that region.
