The magnitude of the electric field at the origin, given an electric potential V(x, y, z), is calculated using the derivatives of the potential with respect to the coordinates. It results in an electric field of approximately 5.83 N/C at the origin.
The student is asking how to calculate the magnitude of the electric field at the origin when given an electric potential function V(x, y, z) = -4 xyz - 3z + 5x. In physics, specifically electromagnetism, the electric field (E) can be derived from the electric potential (V) by taking the negative gradient of V. Thus, the components of the electric field are given by:
Ex = -dV/dx, Ey = -dV/dy, Ez = -dV/dz
For the given potential function, the partial derivatives with respect to x, y, and z at the origin (0,0,0) are:
Ex = -d(-4xyz - 3z + 5x)/dx = -(-3z + 5), Ey = -d(-4xyz)/dy = -(-4xz), Ez = -d(-4xyz - 3z)/dz = -(-4xy - 3)
Substituting x=y=z=0 into these derivatives, we get Ex = 5 N/C, Ey = 0 N/C, and Ez = 3 N/C. Therefore, the magnitude of the electric field at the origin is calculated using:
E = sqrt(Ex² + Ey² + Ez²) = sqrt((5 N/C)² + (0 N/C)² + (3 N/C)²)
The magnitude of the electric field at the origin is approximately 5.83 N/C. It is determined by calculating the vector sum of the electric field components.