Over a certain region of space, the electric potential is V = 5x - 6x2y + 2yz2. Find the expression for the x component of the electric field over this region. (Use x, y, and z as necessary.)
Find the expression for the y component of the electric field over this region.
Find the expression for the z component of the electric field over this region.
What is the magnitude of the field at the point P, which has coordinates (5, 0, -7) m?
To find the expressions for the components of the electric field, we can use the relation:
E = -∇V
where E represents the electric field vector, ∇ is the del operator (used to find the gradient), and V is the electric potential function.
Let's calculate each component of the electric field step-by-step:
1. x-component of the electric field (Ex):
To find Ex, we differentiate the electric potential with respect to x and add the negative sign:
Ex = -dV/dx
Taking the partial derivative of V with respect to x gives:
Ex = -d/dx (5x - 6x^2y + 2yz^2)
= -5 + 12xy
Therefore, the expression for the x-component of the electric field is Ex = -5 + 12xy.
2. y-component of the electric field (Ey):
To find Ey, we differentiate the electric potential with respect to y and add the negative sign:
Ey = -dV/dy
Taking the partial derivative of V with respect to y gives:
Ey = -d/dy (5x - 6x^2y + 2yz^2)
= 6x^2 - 2z^2
Therefore, the expression for the y-component of the electric field is Ey = 6x^2 - 2z^2.
3. z-component of the electric field (Ez):
To find Ez, we differentiate the electric potential with respect to z and add the negative sign:
Ez = -dV/dz
Taking the partial derivative of V with respect to z gives:
Ez = -d/dz (5x - 6x^2y + 2yz^2)
= -2yz
Therefore, the expression for the z-component of the electric field is Ez = -2yz.
Next, let's find the magnitude of the electric field at point P:
Given coordinates of point P: (5, 0, -7) m
To find the magnitude of the electric field at point P, we use the formula:
|E| = sqrt(Ex^2 + Ey^2 + Ez^2)
Substituting the expressions for Ex, Ey, and Ez into the formula, we get:
|E| = sqrt((-5 + 12xy)^2 + (6x^2 - 2z^2)^2 + (-2yz)^2)
Substituting the coordinates of point P into the expression, we have:
|E| = sqrt((-5 + 12(5)(0))^2 + (6(5)^2 - 2(-7)^2)^2 + (-2(5)(-7))^2)
Simplifying the expression:
|E| = sqrt((-5)^2 + (6(5)^2 - 2(-7)^2)^2 + (-2(5)(-7))^2)
|E| = sqrt(25 + (6(25) - 2(49))^2 + (-2(5)(-7))^2)
|E| = sqrt(25 + (150 - 98)^2 + (70)^2)
|E| = sqrt(25 + 52^2 + 70^2)
|E| = sqrt(25 + 2704 + 4900)
|E| = sqrt(7630)
Therefore, the magnitude of the electric field at point P is sqrt(7630) m.