Question

Use \(V=\frac{k q}{r}, E_{x}=-\frac{\partial V}{\partial x}, E_{y}=-\frac{\partial V}{\partial y},\) and \(E_{z}=-\frac{\partial V}{\partial z}\) to derive the expression for the electric field of a point charge, \(q\).

Short answer

Answer: The electric field of a point charge in vector form is given by: \(\vec{E}(x, y, z) = \frac{kq}{(x^2 + y^2 + z^2)^{3/2}}(x\hat{i} + y\hat{j} + z\hat{k})\), where \(k\) is the electrostatic constant, \(q\) is the charge, and \((x, y, z)\) are the coordinates of the point.

Step-by-step solution

Express the potential in Cartesian coordinates

The potential \(V\) is given as \(V = \frac{kq}{r}\), where \(k\) is the electrostatic constant, \(q\) is the charge, and \(r\) is the distance from the charge to the point \((x, y, z)\). We need to express \(r\) in terms of Cartesian coordinates. Using Pythagorean theorem we get: \(r = \sqrt{x^2 + y^2 + z^2}\). So the potential \(V\) becomes: \(V(x, y, z) = \frac{kq}{\sqrt{x^2 + y^2 + z^2}}\).

Calculate partial derivatives

We now have to compute the partial derivatives of the potential \(V\) with respect to coordinates \(x\), \(y\), and \(z\). Let's do that: \(\frac{\partial V}{\partial x} = \frac{-kqx}{(x^2 + y^2 + z^2)^{3/2}}\) \(\frac{\partial V}{\partial y} = \frac{-kqy}{(x^2 + y^2 + z^2)^{3/2}}\) \(\frac{\partial V}{\partial z} = \frac{-kqz}{(x^2 + y^2 + z^2)^{3/2}}\)

Derive the electric field components

By definition, the electric field components are related to the partial derivatives of the potential as follows: \(E_x = -\frac{\partial V}{\partial x}\) \(E_y = -\frac{\partial V}{\partial y}\) \(E_z = -\frac{\partial V}{\partial z}\) Substituting our computed partial derivatives into these definitions, we get: \(E_x = \frac{kqx}{(x^2 + y^2 + z^2)^{3/2}}\) \(E_y = \frac{kqy}{(x^2 + y^2 + z^2)^{3/2}}\) \(E_z = \frac{kqz}{(x^2 + y^2 + z^2)^{3/2}}\)

Express the electric field in vector form

To express the electric field in vector form, we combine the components \(E_x\), \(E_y\), and \(E_z\): \(\vec{E}(x, y, z) = E_x\hat{i} + E_y\hat{j} + E_z\hat{k} = \frac{kq}{(x^2 + y^2 + z^2)^{3/2}}(x\hat{i} + y\hat{j} + z\hat{k})\) This equation represents the electric field of a point charge \(q\) at the coordinates \((x, y, z)\) in vector form.