Question

A charge \(Q_{0}\) located at the origin in free space produces a field for which \(E_{z}=1 \mathrm{kV} / \mathrm{m}\) at point \(P(-2,1,-1) .(a)\) Find \(Q_{0} .\) Find \(\mathbf{E}\) at \(M(1,6,5)\) in (b) rectangular coordinates; ( \(c\) ) cylindrical coordinates; \((d)\) spherical coordinates.

Short answer

Answer: The electric field components of point M are: 1. Rectangular coordinates: \(-0.185\textbf{i} -1.109\textbf{j} -0.926\textbf{k}\, \mathrm{kV/m}\) 2. Cylindrical coordinates: \(-0.227\textbf{e}_\rho + 0.170\textbf{e}_\phi - 0.926\textbf{e}_z\, \mathrm{kV/m}\) 3. Spherical coordinates: \(-0.467\textbf{e}_r + 0.694\textbf{e}_\theta + 0.170\textbf{e}_\phi\, \mathrm{kV/m}\)

Step-by-step solution

Find Q_0 using given E_z at point P

To determine the charge Q_0 responsible for the electric field, we will first need to find the position vector (r_P) at point P. Using the given coordinates of point P(-2, 1, -1), we find r_P: \( \textbf{r}_P = (-2\textbf{i} + 1\textbf{j} - 1\textbf{k})\). Now, find the magnitude of the position vector: \( | \textbf{r}_P | = \sqrt{(-2)^2 +1^2 + (-1)^2} = \sqrt{6}\). Given that \(E_z\) = 1 kV/m, we can write: \( E_{zP} = \dfrac{Q_0}{4 \pi \epsilon _0 |\textbf{r}_P|^3} \cdot (-1\textbf{k})\). Solving for Q_0 and multiplying both sides by \(4 \pi \epsilon _0 |\textbf{r}_P|^3\): \( Q_0 = E_{zP} \cdot 4 \pi \epsilon _0 |\textbf{r}_P|^3 \cdot (-1\textbf{k})\). Using ε_0 = 8.85 x 10^{-12} C^2/Nm^2 and E_z = 1000 V/m, we get: \( Q_0 = (1000 \mathrm{V/m}) \cdot 4 \pi (8.85 \times 10^{-12} \mathrm{C}^2/\mathrm{N}\mathrm{m}^2) | \textbf{r}_P|^3 \cdot (-1\textbf{k})\). \( Q_0 = -6.73 \times 10^{-9} \, \mathrm{C} \cdot \mathrm{k}\).

Find the position vector for point M

To determine the position vector for point M with coordinates (1, 6, 5), we have: \( \textbf{r}_M = (1\textbf{i} + 6\textbf{j} + 5\textbf{k})\). Find the magnitude of r_M: \(|\textbf{r}_M| = \sqrt{1^2 + 6^2 + 5^2} = \sqrt{62}\).

Find the electric field components in rectangular coordinates

Now, we can find the electric field components at point M using the general formula of the electric field produced by a point charge: \( \textbf{E}_M = \dfrac{Q_0}{4 \pi \epsilon _0 |\textbf{r}_M|^3} \textbf{r}_M\). Plugging in the values for Q_0, ε_0, and r_M, we get: \( \textbf{E}_M = \dfrac{-6.73 \times 10^{-9} \mathrm{C} \cdot \textbf{r}_M}{ 4 \pi (8.85 \times 10^{-12} \mathrm{C}^2/\mathrm{N}\mathrm{m}^2) (\sqrt{62})^3}\). Calculate the electric field components: \( \textbf{E}_M = (-0.185\textbf{i} -1.109\textbf{j} -0.926\textbf{k})\, \mathrm{kV/m}\).

Convert to cylindrical coordinates

To convert the rectangular coordinates to cylindrical coordinates, we use the following conversions: ρ = √(x^2 + y^2) ϕ = arctan(y/x) z = z Let's find ρ and ϕ for point M: ρ = √(1^2 + 6^2) = √37 ϕ = arctan(6/1) = 80.54° Using these values and applying the transformations, we find the cylindrical components field: \( \textbf{E}_M^{cylindrical} = (E_{\rho} \textbf{e}_\rho + E_{\phi} \textbf{e}_\phi + E_{z} \textbf{e}_z)\). \( \textbf{E}_M^{cylindrical} \approx (-0.227\textbf{e}_\rho + 0.170\textbf{e}_\phi - 0.926\textbf{e}_z)\, \mathrm{kV/m}\).

Convert to spherical coordinates

To convert the rectangular coordinates to spherical coordinates, we use the following conversions: r = √(x^2 + y^2 + z^2) θ = arccos(z/r) ϕ = arctan(y/x) We have already found r and ϕ for point M, now let's find θ: θ = arccos(5/√62) = 50.68° Using these values and applying the transformations, we find the spherical components field: \( \textbf{E}_M^{spherical} = (E_{r} \textbf{e}_r + E_{\theta} \textbf{e}_\theta + E_{\phi} \textbf{e}_\phi)\). \( \textbf{E}_M^{spherical} \approx (-0.467\textbf{e}_r + 0.694\textbf{e}_\theta + 0.170\textbf{e}_\phi)\, \mathrm{kV/m}\).