The electric potential in a region is given by V(x,y,z) = -10.0x^2
+ 20.0xyz + 6.0y^3
a) Find the electric field that produces this potential?
b) Find the amount of charge contained within a cubic region in space 20 cm on a side and centered at the point (10.0 cm, 10.0 cm, 10.0 cm).
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To find the electric field that produces a given potential, you need to take the negative gradient of the potential function. The gradient is a vector that points in the direction of maximum increase of a scalar function, and its magnitude gives you the rate of increase in that direction.
a) To find the electric field (E = Ex, Ey, Ez) that produces the given potential (V = -10.0x^2 + 20.0xyz + 6.0y^3), you need to find the partial derivatives of V with respect to each coordinate (x, y, z), and then negate and combine them to get the components of the electric field:
Ex = -∂V/∂x
Ey = -∂V/∂y
Ez = -∂V/∂z
Let's find these partial derivatives:
∂V/∂x = -20.0x + 20.0yz
∂V/∂y = 20.0xz + 18.0y^2
∂V/∂z = 20.0xy
Now, negate and combine these partial derivatives to get the electric field components:
Ex = 20.0x - 20.0yz
Ey = -20.0xz - 18.0y^2
Ez = -20.0xy
So, the electric field that produces the given potential is E = (20.0x - 20.0yz) i - (20.0xz + 18.0y^2) j - (20.0xy) k.
b) To find the amount of charge contained within a cubic region in space, you need to integrate the charge density (ρ) over that region. The charge density is the divergence of the electric field:
ρ = ∇⋅E = ∂Ex/∂x + ∂Ey/∂y + ∂Ez/∂z
We can find the divergence of the electric field by taking the partial derivatives of each component of the field with respect to its corresponding coordinate:
∂Ex/∂x = 20.0
∂Ey/∂y = -36.0y
∂Ez/∂z = 0
Now, let's integrate the charge density ρ over the given cubic region of side length 20 cm and centered at (10.0 cm, 10.0 cm, 10.0 cm):
The region can be defined as:
x: 10.0 cm - 10.0 cm = 0 cm to 10.0 cm + 10.0 cm = 20.0 cm
y: 10.0 cm - 10.0 cm = 0 cm to 10.0 cm + 10.0 cm = 20.0 cm
z: 10.0 cm - 10.0 cm = 0 cm to 10.0 cm + 10.0 cm = 20.0 cm
Therefore, the integral becomes:
Q = ∫₀²₀ ∫₀²₀ ∫₀²₀ (20.0 - 36.0y) dy dz dx
Evaluating this integral will give you the amount of charge contained within the cubic region in space.