Answer:
a) An equation for the x-component of the electric field.
Eₓ = (-15xy³ + 5.32xy⁴z²) N/C
b) An equation for the y-component of the electric field.
Eᵧ = (-22.5x²y² + 10.64x²y³z²) N/C
c) An equation for the z-component of the electric field.
Ez = (5.32x²y⁴z) N/C
d) At (-5.0, 2.0, 1.5) m, the electric field is given as
E = (-357.6î + 2,538ĵ + 3,192ķ) N/C
Magnitude of the electric field = 4,093.7 N/C
Step-by-step explanation:
The electric field is given by the negative of the gradient of the electric potential,
E = −grad V
E = - ∇V
The electric potential is given as
V(x,y,z) = 3αx²y³ - 2γx²y⁴z²
α = 2.5 V/m⁵ and γ = 1.33 V/m⁸
V(x,y,z) = 7.5x²y³ - 2.66x²y⁴z²
grad = ∇ = (∂/∂x)î + (∂/∂y)ĵ + (∂/∂z)ķ
E = -grad V = -∇V
= -[(∂V/∂x)î + (∂V/∂y)ĵ + (∂V/∂z)ķ
E = -(∂V/∂x)î - (∂V/∂y)ĵ - (∂V/∂z)ķ
E = Eₓî + Eᵧĵ + Ez ķ
a) An equation for the x-component of the electric field.
Eₓ = -(∂V/∂x) = -(∂/∂x)(V)
= -(∂/∂x)(7.5x²y³ - 2.66x²y⁴z²)
= -(15xy³ - 5.32xy⁴z²)
= (-15xy³ + 5.32xy⁴z²)
b) An equation for the y-component of the electric field.
Eᵧ = -(∂V/∂y) = -(∂/∂x)(V)
= -(∂/∂y)(7.5x²y³ - 2.66x²y⁴z²)
= -(22.5x²y² - 10.64x²y³z²)
= (-22.5x²y² + 10.64x²y³z²)
c) An equation for the z-component of the electric field.
Ez = -(∂V/∂z) = -(∂/∂x)(V)
= -(∂/∂z)(7.5x²y³ - 2.66x²y⁴z²)
= -(0 - 5.32x²y⁴z)
= (5.32x²y⁴z)
d) E = Eₓî + Eᵧĵ + Ez ķ
E = (-15xy³ + 5.32xy⁴z²)î + (-22.5x²y² + 10.64x²y³z²)ĵ + (5.32x²y⁴z) ķ
At (-5.0, 2.0, 1.5) m
x = -5 m
y = 2 m
z = 1.5 m
Eₓ = (-15xy³ + 5.32xy⁴z²)
= (-15×-5×2³) + (5.32×-5×2⁴×1.5²)
= 600 - 957.6 = -357.6
Eᵧ = (-22.5x²y² + 10.64x²y³z²)
= (-22.5×(-5)²×2²) + (10.64×(-5)²×2³×1.5²)
= -2250 + 4788 = 2538
Ez = (5.32x²y⁴z) = (5.32×(-5)²×2⁴×1.5)
= 3192
E = -357.6î + 2,538ĵ + 3,192ķ
Magnitude = /E/ = √[(-357.6)² + 2538² + 3192²]
= 4,093.6763135353 = 4,093.7 N/C
Hope this Helps!!!!