Question

If \(\vec E = - \left( {2{y^3} - 3y{z^2}} \right)\hat x - \left( {6x{y^2} - 3x{z^2}} \right)\hat y + \left( {6xyz} \right)\hat z\) is the electric field in a source free region, a valid expression for the electrostatic potential is
1. xy³ – yz²
2. 2xy³ – xyz²
3. y³ + xyz²
4. 2xy³ – 3xyz²

Answer 1

Correct Answer - Option 4 : 2xy³ – 3xyz²

Concept:

The electric field is related to the potential as:

\(\vec E = - ∇ V\)

\(∇ =\frac{\partial}{\partial x}\hat a_x+\frac{\partial}{\partial y}\hat a_y+\frac{\partial}{\partial z}\hat a_z\)

Calculation:

We can verify all options with the Maxwell equation \(\vec E = - ∇ V\)

Option 1: V = xy3 – yz2

\( - ∇ V=-(\frac{\partial}{\partial x} (xy^3 – yz^2)\hat a_x+\frac{\partial}{\partial y}(xy^3 – yz^2)\hat a_y+\frac{\partial}{\partial z}(xy^3 – yz^2)\hat a_z)\)

= -y³ a_x - (3xy² - z²) a_y - 2yz a_z

Since this is not equal to the given Electric field, this cannot be a valid expression for the given electrostatic potential.

Option 2: V = 2xy3 – xyz2

\( - ∇ V=-(\frac{\partial}{\partial x} (2xy^3 – xyz^2)\hat a_x+\frac{\partial}{\partial y}(2xy^3 – xyz^2)\hat a_y+\frac{\partial}{\partial z}(2xy^3 – xyz^2)\hat a_z)\)

= -[(2y3 - yz²) ax + (6xy2 - xz2) ay - 2xyz az]

Since this is not equal to the given Electric field, this cannot be a valid expression for the given electrostatic potential.

Option 3: V = y3 + xyz2

\( - ∇ V=-(\frac{\partial}{\partial x} (y^3 + xyz^2)\hat a_x+\frac{\partial}{\partial y}(y^3 + xyz^2)\hat a_y+\frac{\partial}{\partial z}(y^3 + xyz^2)\hat a_z)\)

= -[yz2 ax + (3y2 + xz2) ay + 2xyz az]

Since this is not equal to the given Electric field, this cannot be a valid expression for the given electrostatic potential.

Option 4: V = 2xy3 – 3xyz2

\( - ∇ V=-(\frac{\partial}{\partial x} (2xy^3 – 3xyz^2)\hat a_x+\frac{\partial}{\partial y}(2xy^3 – 3xyz^2)\hat a_y+\frac{\partial}{\partial z}(2xy^3 – 3xyz^2)\hat a_z)\)

= -[(2y³ - 3yz²)ax + (6xy2 - 3xz2) ay - 6xyz az]

= -(2y3 - 3yz2)ax - (6xy2 - 3xz2) ay + 6xyz az

Since E = - ∇ V for this case, this this is a valid expression for the given electrostatic potential.