Q16Engineering Physics

Question

If a potential function is given by the expression, $ϕ = x y z$ , determine the potential gradient and also prove that the vector is irrotational.

Answer 1

In electromagnetic theory and fluid dynamics, vector fields are rigorously analyzed using the vector operator "Del" ( $\nabla$ ). Two fundamental operations are the Gradient (which turns a scalar field into a vector field indicating the steepest slope) and the Curl (which measures the microscopic rotation of a vector field). A profound theorem in vector calculus states that the curl of the gradient of any scalar field is always identically zero ( $\nabla \times (\nabla ϕ) = 0$ ). We will prove this using the given scalar potential.

1. Calculation of the Gradient Vector Field ( $V$ )

We are given a scalar potential field: $ϕ (x, y, z) = x y z$ . The gradient operation ( $\nabla ϕ$ ) takes partial derivatives of this scalar function with respect to each spatial coordinate and assigns them to the respective Cartesian unit vectors ( $\hat{i}, \hat{j}, \hat{k}$ ).

V = \nabla ϕ = \frac{\partial ϕ}{\partial x} \hat{i} + \frac{\partial ϕ}{\partial y} \hat{j} + \frac{\partial ϕ}{\partial z} \hat{k}

Taking the partial derivatives:

$\frac{\partial}{\partial x} (x y z) = y z$ (treating $y$ and $z$ as constants)
$\frac{\partial}{\partial y} (x y z) = x z$ (treating $x$ and $z$ as constants)
$\frac{\partial}{\partial z} (x y z) = x y$ (treating $x$ and $y$ as constants)

Substituting these back yields the resultant vector field:

V = (y z) \hat{i} + (x z) \hat{j} + (x y) \hat{k}

2. Proof of Irrotationality via the Curl

A vector field is strictly defined as irrotational (or conservative) if it has absolutely no circulation or "churning" at any point in space. Mathematically, this condition is met if and only if the curl of the vector field is exactly equal to the zero vector ( $\nabla \times V = 0$ ). We evaluate the curl by computing the determinant of a $3 \times 3$ matrix:

\nabla \times V = \hat{i} \frac{\partial}{\partial x} V_{x} \hat{j} \frac{\partial}{\partial y} V_{y} \hat{k} \frac{\partial}{\partial z} V_{z} = \hat{i} \frac{\partial}{\partial x} y z \hat{j} \frac{\partial}{\partial y} x z \hat{k} \frac{\partial}{\partial z} x y

Expanding the determinant across the first row:

\nabla \times V = \hat{i} (\frac{\partial ( x y )}{\partial y} - \frac{\partial ( x z )}{\partial z}) - \hat{j} (\frac{\partial ( x y )}{\partial x} - \frac{\partial ( y z )}{\partial z}) + \hat{k} (\frac{\partial ( x z )}{\partial x} - \frac{\partial ( y z )}{\partial y})

Evaluating the inner partial derivatives carefully:

\nabla \times V = \hat{i} (x - x) - \hat{j} (y - y) + \hat{k} (z - z)

\nabla \times V = 0 \hat{i} - 0 \hat{j} + 0 \hat{k} = 0

Conclusion: Because the curl evaluates exactly to the zero vector at all points $(x, y, z)$ , we have successfully proven that the vector field $V$ is undeniably irrotational. This verifies the physical principle that any field derived directly from a scalar potential is a conservative field (like the electrostatic field derived from voltage).

Q16Engineering Physics

Question

Answer

1. Calculation of the Gradient Vector Field (V)

2. Proof of Irrotationality via the Curl

1. Calculation of the Gradient Vector Field ( $V$ )