The magnitude of the electric field at the point (-2, 2, 0) is 46.52 V/m
<h3>How to calculate the electric field?</h3>
Since in a certain region of space, the potential is given by v = 2x − 5x²y 3yz²,
The electric field, E = -grad(V) where grad(V) = (dV/dx)i + (dV/dy)j + (dV/dz)k
Since V = 2x − 5x²y + 3yz²
dV/dx = d(2x − 5x²y + 3yz²)/dx
= d2x/dx - d5x²y/dx + d3yz²/dx
= 2 - 10xy + 0
= 2 - 10xy
dV/dy = d(2x − 5x²y + 3yz²)/dy
= d2x/dy - d5x²y/dy + d3yz²/dy
= 0 - 5x² + 3z²
= - 5x² + 3z²
dV/dz = d(2x − 5x²y + 3yz²)/dz
= d2x/dz - d5x²y/dz + d3yz²/dz
= 0 - 0 + 6z
= 6z
<h3>
The electric field</h3>
So, E = -grad(V)
= -[(dV/dx)i + (dV/dy)j + (dV/dz)k]
= -[(2 - 10xy)i + (- 5x² + 3z²)j + (6z)k]
So, the electric field at (-2, 2, 0) is
E = -[(2 - 10xy)i + (- 5x² + 3z²)j + (6z)k]
E = -[(2 - 10(-2)(2))i + (- 5(2)² + 3(0)²)j + (6(0))k]
E = -[(2 + 40)i + (- 5(4) + 0)j + (0)k]
E = -[42i + (-20 + 0)j + (0)k]
E = -[42i - 20j + 0k]
E = -42i + 20j - 0k
<h3>The magnitude of the electric field</h3>
So, the magnitude of E at (-2, 2, 0) is
|E| = √[(-42)² + 20² + 0²]
= √[1764 + 400 + 0]
= √2164
= 46.52 V/m
So, the magnitude of the electric field at the point (-2, 2, 0) is 46.52 V/m
Learn more about electric field here:
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then vertex is max value
then vertex is min value
