1) Potential difference: 144.6 V
2) Potential difference: -909 V
Explanation:
1)
Given an electric field in three dimensions, with components [tex]E_x,E_y,E_z[/tex], we have that each component of the field is related to the electric potential V by:
[tex]E_x=-\frac{dV}{dx}\\E_y=-\frac{dV}{dy}\\E_z=-\frac{dV}{dz}[/tex]
For the field in this problem, we know that
[tex]E_z=0[/tex]
Which means that
[tex]dE_z=-\frac{dV}{dz}\rightarrow dV=-dE_z dz=0[/tex]
and so the potential does not depend on z, so we can re-write it as [tex]V(x,y)[/tex].
Then we take the equation in x:
[tex]E_x=6x^2y=-\frac{dV}{dx}[/tex]
Solving for V,
[tex]V=-\int 6x^2 ydx =-6\frac{x^3}{3}y+f(y)=-2x^3y+f(y)[/tex]
where f(y) is a function depending on y.
From the equation in y,
[tex]E_y=-\frac{dV}{dy}[/tex]
And substituting the expression of V,
[tex]E_y=-\frac{d}{dy}(-2x^3y-f(y))=2x^3-f'(y)[/tex]
We know also that
[tex]E_y=2x^3+2y[/tex]
And by comparing these last 2 equations, we infer that
[tex]f'(y)=-2y\\f(y)=-\int 2ydy=-y^2+C[/tex]
So, the expression for the electric potential is
[tex]V(x,y)=-2x^3y-y^2+C[/tex]
where C is a constant.
Now we can find the electrical potential difference between the origin and the other point.
The origin has coordinates (0,0,0), so the potential is
[tex]V(0,0)=-2\cdot 0^3(0)-0^2+C=C[/tex] (V)
The other point has coordinates (-2.7 m,4.1 m,0.0 m), so the potential is
[tex]V(-2.7,4.1)=-2(-2.7)^3(4.1)-(4.1)^2+C=144.6+C[/tex] (V)
So, the potential difference is
[tex]\Delta V = 144.6 +C - C = 144.6 V[/tex]
2)
In this second case, we want to find the potential difference between the origin and another point.
The potential is
[tex]V(x,y)=-2x^3y-y^2+C[/tex]
The origin has coordinates (0,0,0), so the potential is
[tex]V(0,0)=-2\cdot 0^3(0)-0^2+C=C[/tex] (V)
The second point has coordinates (4.2 m, 5.9 m, 0.0 m), so the potential there is
[tex]V(4.2,5.9)=-2(4.2)^3(5.9)-(5.9)^2+C=-909 +C[/tex] (V)
Therefore, the potential difference is given by the difference of these two values:
[tex]\Delta V=-909 +C -C = -909 V[/tex]
Learn more about electric field and potential:
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