Question

The electric potential in a certain region is given by the equation V(x,y,z) = 3αx2y3 - 2γx2y4z2 where the potential is in volts when the positions are given in meters.

The constants in this equation are α = 5.1 V/m5 and γ = 0.24 V/m8. Using this information, calculate the magnitude of the electric field at the point (x1,y1,z1) = (-5.0, 2.0, 1.5) m in units of newtons per coulomb.
E = ? N/C

Answer 1

The electric field is calculated by taking the negative gradient of the electric potential function, and the magnitude at a point is obtained by summing the squares of the component fields and taking the square root.

Step-by-step explanation:

The question involves the concept of calculating the electric field (E) from the electric potential (V). The electric potential in the given region is expressed as V(x, y, z) = 3αx²y³ - 2γx²y´z², with the constants α = 5.1 V/m⁵ and γ = 0.24 V/m⁸.

To find the magnitude of the electric field at the point (-5.0, 2.0, 1.5) m, one must use the negative of the gradient of the potential:

E = -∇V

The components of the electric field are as follows:

Ex = - ∂V/∂x

Ey = - ∂V/∂y

Ez = - ∂V/∂z

After calculating the partial derivatives and evaluating at the given point, we would sum the square of each component, then take the square root to find the electric field magnitude:

|E| = √(Ex² + Ey² + Ez²)

This process involves calculus and vector analysis, crucial parts of advanced physics courses.