Question

A conducting solid sphere of radius \(20.0 \mathrm{~cm}\) is located with its center at the origin of a three-dimensional coordinate system. A charge of \(0.271 \mathrm{nC}\) is placed on the sphere. a) What is the magnitude of the electric field at point $$ (x, y, z)=(23.1 \mathrm{~cm}, 1.10 \mathrm{~cm}, 0.00 \mathrm{~cm}) ? $$ b) What is the angle of this electric field with the \(x\) -axis at this point? c) What is the magnitude of the electric field at point \((x, y, z)=(4.10 \mathrm{~cm}, 1.10 \mathrm{~cm}, 0.00 \mathrm{~cm}) ?\)

Short answer

b) What is the angle of the electric field at point (23.1 cm, 1.10 cm, 0.00 cm) with the x-axis? c) What is the electric field at point (4.10 cm, 1.10 cm, 0.00 cm)?

Step-by-step solution

a) Electric field at point(x, y, z)=(23.1cm, 1.10cm, 0.00cm)

: To calculate the electric field at \((23.1\mathrm{~cm}, 1.10\mathrm{~cm}, 0\mathrm{~cm})\), we first need to convert the coordinates to meters. The coordinates in meters are \((0.231\mathrm{~m}, 0.011\mathrm{~m}, 0\mathrm{~m})\). The formula for the electric field from a point charge \(Q\) at a distance \(r\) is given by: $$ E = \frac{kQ}{r^2} $$ Where $$ E = \text{electric field}\newline k = 8.99 × 10^9 \mathrm{Nm^2C^{-2}} \;\text{(Coulomb's constant)}\newline Q = 0.271 × 10^{-9} \mathrm{C}\;(\text{Charge on the sphere})\newline r^2 = (0.231\mathrm{~m})^2 + (0.011\mathrm{~m})^2 $$ Calculate \(r^2\) and then plug in the values into the equation to find the electric field.

b) Angle with x-axis

: Now, we have to find the angle of the electric field with the x-axis. We can use the trigonometric relationship \(\tan^{-1}\theta = \frac{y}{x}\). $$ \theta = \tan^{-1}\left( \frac{0.011}{0.231} \right) $$ Calculate the angle θ to find the angle of the electric field with the x-axis.

c) Electric field at point (x, y, z)=(4.10 cm, 1.10 cm, 0.00 cm)

: Now, we will determine the electric field at \((4.10\mathrm{~cm}, 1.10\mathrm{~cm}, 0\mathrm{~cm})\). First, we have to convert the coordinates to meters. The coordinates in meters are \((0.041\mathrm{~m}, 0.011\mathrm{~m}, 0\mathrm{~m})\). The distance from the sphere center to this point is less than the radius, so we can use the following equation for electric fields inside a conducting solid sphere: $$ E = \frac{kQr}{R^3} $$ Where $$ E = \text{electric field}\newline k = 8.99 × 10^9 \mathrm{Nm^2C^{-2}} \;\text{(Coulomb's constant)}\newline Q = 0.271 × 10^{-9} \mathrm{C}\;(\text{Charge on the sphere})\newline R = 0.20 \mathrm{m}\;(\text{radius of the sphere})\newline r^2 = (0.041\mathrm{~m})^2 + (0.011\mathrm{~m})^2 $$ Calculate \(r^2\) and then plug in the values into the equation to find the electric field at point \((4.10\mathrm{~cm}, 1.10\mathrm{~cm}, 0\mathrm{~cm})\).