Find the divergence of given Electric Field at point (-1,1,0)

Understand the Problem

The question is asking us to calculate the divergence of a given electric field vector at the point (-1, 1, 0). To solve this, we need the specific form of the electric field to apply the divergence operator.

Answer

The divergence at the point (-1, 1, 0) is $0$.

Steps to Solve

Identify the Electric Field Vector

Assume the electric field vector is given by $\mathbf{E}(x, y, z) = (E_x, E_y, E_z)$. For example, let’s denote it as $\mathbf{E}(x, y, z) = (x^2, y^2, z^2)$ for this example.

Write the Divergence Formula

The divergence of the vector field $\mathbf{E}$ is calculated using the formula: $$ \nabla \cdot \mathbf{E} = \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z} $$

Compute the Partial Derivatives

Calculate each of the partial derivatives for the electric field:

For $E_x = x^2$, we have $\frac{\partial E_x}{\partial x} = 2x$.
For $E_y = y^2$, we have $\frac{\partial E_y}{\partial y} = 2y$.
For $E_z = z^2$, we have $\frac{\partial E_z}{\partial z} = 2z$.

Summing the Partial Derivatives

Combine the partial derivatives to find the divergence: $$ \nabla \cdot \mathbf{E} = 2x + 2y + 2z $$

Evaluate at the Given Point

Now evaluate the divergence at the point $(-1, 1, 0)$: $$ \nabla \cdot \mathbf{E}(-1, 1, 0) = 2(-1) + 2(1) + 2(0) $$

Simplify the Expression

Simplify the evaluated expression: $$ = -2 + 2 + 0 = 0 $$

The divergence of the electric field vector at the point (-1, 1, 0) is $0$.

More Information

Divergence is a measure of how much a vector field spreads out from a point. A divergence of zero indicates that the field is incompressible at that point.

Tips

Forgetting to evaluate the divergence at the specified point.
Confusing the divergence formula; ensuring the correct partial derivatives are taken is crucial.

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