The electric potential in a certain region is given by the equation V(x,y,z) = 3αx^2y^3 - 2γx^2y^4z^2 where the potential is in volts when the positions are given in meters. The constants in this equation are α = 3.6 V/m^5 and γ = 1.34 V/m^8
Write an equation for the x-component of the electric field.
Write an equation for the y-component of the electric field.
Write an equation for the z-component of the electric field.
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.
To find the x-component of the electric field (Ex), we need to take the derivative of the electric potential V(x, y, z) with respect to x:
Ex = -∂V/∂x
Given V(x, y, z) = 3αx^2y^3 - 2γx^2y^4z^2,
Let's calculate the derivative of V(x, y, z) with respect to x:
∂V/∂x = 6αxy^3 - 4γxy^4z^2
So, the equation for the x-component of the electric field is:
Ex = 6αxy^3 - 4γxy^4z^2
To find the y-component of the electric field (Ey), we need to take the derivative of the electric potential V(x, y, z) with respect to y:
Ey = -∂V/∂y
Given V(x, y, z) = 3αx^2y^3 - 2γx^2y^4z^2,
Let's calculate the derivative of V(x, y, z) with respect to y:
∂V/∂y = 9αx^2y^2 - 8γx^2y^3z^2
So, the equation for the y-component of the electric field is:
Ey = 9αx^2y^2 - 8γx^2y^3z^2
To find the z-component of the electric field (Ez), we need to take the derivative of the electric potential V(x, y, z) with respect to z:
Ez = -∂V/∂z
Given V(x, y, z) = 3αx^2y^3 - 2γx^2y^4z^2,
Let's calculate the derivative of V(x, y, z) with respect to z:
∂V/∂z = -4γx^2y^4z
So, the equation for the z-component of the electric field is:
Ez = -4γx^2y^4z
To 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, we need to use the equation:
|E| = √(Ex^2 + Ey^2 + Ez^2)
Let's substitute the given values into the equations for Ex, Ey, and Ez and calculate the magnitude:
Ex = 6αxy^3 - 4γxy^4z^2
= 6(3.6 V/m^5)(-5.0 m)(2.0 m)^3 - 4(1.34 V/m^8)(-5.0 m)(2.0 m)^4(1.5 m)^2
Ey = 9αx^2y^2 - 8γx^2y^3z^2
= 9(3.6 V/m^5)(-5.0 m)^2(2.0 m)^2 - 8(1.34 V/m^8)(-5.0 m)^2(2.0 m)^3(1.5 m)^2
Ez = -4γx^2y^4z
= -4(1.34 V/m^8)(-5.0 m)^2(2.0 m)^4(1.5 m)
Now, substitute these values into the magnitude equation:
|E| = √(Ex^2 + Ey^2 + Ez^2)
= √[(Ex)^2 + (Ey)^2 + (Ez)^2]
This will give you the magnitude of the electric field at the given point.