In a certain region of space the electric potential is V= 5x-3x^2y+2yz^2. Find the expressions for the x,y,z compenents of the electric field in this region. What is the magnitude of the field at the point P, which has coordinates 1,0,-2)m?
I don't know how to start this problem..is there a formula to use? Any help would be greatly appreciated!
Oh, electric potential and electric field, it's like a shocking combination! Alright, let's zap into action and solve this for you! To find the components of the electric field (Ex, Ey, Ez), we can use the following formula:
Ex = -∂V/∂x
Ey = -∂V/∂y
Ez = -∂V/∂z
Now, let's differentiate the given electric potential V with respect to each variable separately:
∂V/∂x = 5 - 6xy
∂V/∂y = -3x^2 + 2z^2
∂V/∂z = 2yz
Plugging in the point P with coordinates (1, 0, -2) into these expressions, we get:
Ex = -∂V/∂x = -5 - 6(1)(0) = -5
Ey = -∂V/∂y = -3(1)^2 + 2(-2)^2 = -3 + 8 = 5
Ez = -∂V/∂z = 2(1)(-2) = -4
The components of the electric field at point P are Ex = -5, Ey = 5, and Ez = -4. But wait, there's more! To find the magnitude of the electric field at point P, we can use the magnitude formula:
|E| = √(Ex^2 + Ey^2 + Ez^2)
Plugging in the values we found earlier, we get:
|E| = √((-5)^2 + (5)^2 + (-4)^2)
= √(25 + 25 + 16)
= √66
So, the magnitude of the electric field at point P is √66. Phew, that was electrifying! Hope that helps!
To find the expressions for the x, y, and z components of the electric field, we can make use of Coulomb's law:
E = -∇V
where E is the electric field vector, V is the electric potential, and ∇ is the gradient operator.
To calculate each component of the electric field, we differentiate the potential with respect to each variable. Let's start with the x-component:
Ex = -∂V/∂x
Differentiating V with respect to x, we get:
∂V/∂x = 5 - 6xy
Therefore, the x-component of the electric field is:
Ex = -(5 - 6xy)
Similarly, we can find the y-component:
Ey = -∂V/∂y
Differentiating V with respect to y, we get:
∂V/∂y = -3x^2 + 2z^2
Therefore, the y-component of the electric field is:
Ey = -(-3x^2 + 2z^2)
Which simplifies to:
Ey = 3x^2 - 2z^2
Finally, let's find the z-component:
Ez = -∂V/∂z
Differentiating V with respect to z, we get:
∂V/∂z = 2yz
Therefore, the z-component of the electric field is:
Ez = -2yz
To find the magnitude of the field at point P(1, 0, -2), we substitute the values into our expressions for Ex, Ey, and Ez:
Ex = -(5 - 6xy) = -(5 - 6(1)(0)) = -5
Ey = 3x^2 - 2z^2 = 3(1)^2 - 2(-2)^2 = 3 - 8 = -5
Ez = -2yz = -2(0)(-2) = 0
The magnitude of the electric field at point P is given by:
|E| = sqrt(Ex^2 + Ey^2 + Ez^2) = sqrt((-5)^2 + (-5)^2 + 0^2) = sqrt(50) ≈ 7.071
To find the expressions for the components of the electric field (Ex, Ey, Ez) in this region, you can use the relationship between electric field and electric potential:
E = -∇V
where ∇ is the gradient operator (del operator) and V is the electric potential. The components of the electric field can be found by taking the partial derivatives of the potential with respect to each coordinate.
To start, let's find Ex, the x-component of the electric field:
Ex = -∂V/∂x
To find ∂V/∂x, differentiate V with respect to x while treating y and z as constants:
∂V/∂x = 5 - 6xy
Next, let's find Ey, the y-component of the electric field:
Ey = -∂V/∂y
To find ∂V/∂y, differentiate V with respect to y while treating x and z as constants:
∂V/∂y = -3x^2 + 2z^2
Finally, let's find Ez, the z-component of the electric field:
Ez = -∂V/∂z
To find ∂V/∂z, differentiate V with respect to z while treating x and y as constants:
∂V/∂z = 2yz
Now that we have the expressions for Ex, Ey, and Ez, we can calculate the magnitude of the electric field (E) at the point P (1, 0, -2).
To calculate the magnitude of the electric field at point P, use the formula:
E = sqrt(Ex^2 + Ey^2 + Ez^2)
Substitute the given values into the expressions for Ex, Ey, and Ez, and calculate E.
The E field in direction x is
Ex = -dV/dx (partial deriviative)
and similarly for Ey and Ez
Ey = -dV/dy
Ez = -dV/dz = -4yz
Do the other two partial derivatives, and then plug in the appropriate x,y,z values for that point.
As the final step,
|E| = sqrt(Ex^2+Ey^2+Ez^2)