The electric potential in a certain region is given by the equation V(x, y, z) =

Question

The electric potential in a certain region is given by the equation V(x, y, z) = 3alphax^2y^3 - 2yx^2y^4z^2 where the potential is in volts when the positions are given in meters. The constants in this equation are given below. Randomized Variables a = 4.9 Vim^5 gamma = 0.91 V/m^8 A 25% Part (b) Write an equation for the y-component of the electric field. H A 25% Part (c) Write an equation for the z-component of the electric field. A 25% Part (d) Calculate the magnitude of the electric field at the point (x_1, y_1 z_j) = (-5.0, 2.0, 1.5) m in units of newtons per coulomb.

Explanation / Answer

Here ,

V = 3 a * x^2 y^3 - 2 gamma * x^2 y^4 * z^2

V = 14.7 * x^2 y^3 - 1.82 * x^2 y^4 * z^2

a) for Ex

Ex = - dV/dx i

Ex = - d/dx(14.7 * x^2 y^3 - 1.82 * x^2 y^4 * z^2)

Ex = - 29.4 x y^3 + 3.64 x y^4 z^2

b)

Ey = - dV/dy j

Ey = - d/dy(14.7 * x^2 y^3 - 1.82 * x^2 y^4 * z^2)

Ey = -44.1 x^2 y^2 + 7.82 x^2 y^3 z^2

c)

Ez = - dV/dz k

Ez = - d/dz(14.7 * x^2 y^3 - 1.82 * x^2 y^4 * z^2)

Ez = 3.64 x y^4 z^2

d) at (-5 , 2 ,1.5)

Ex = - 29.4 (-5) *2^3 + 3.64 (-5) *2^4 *1.5^2

Ex = 520.8 V/m

Ey = -44.1 (-5)^2*2^2 + 7.82 (-5)^2 *(2)^3 *(1.5)^2

Ey = -891 V/m

Ez = 3.64 * (-5) * (2)^4 * (1.5)^2

Ez = -655.2 V/m

magnitude of electric field = sqrt(520.8^2 + 891^2 + 655.2^2)

magnitude of electric field = 1222.5 V/m

the magnitude of electric field is 1222.5 V/m

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