Problem 47 - Fundamentals of Physics

Question

An unknown charge sits on a conducting solid sphere of radius \(10 \mathrm{~cm}\). If the electric field \(15 \mathrm{~cm}\) from the center of the sphere has the magnitude \(3.0 \times 10^{3} \mathrm{~N} / \mathrm{C}\) and is directed radially inward, what is the net charge on the sphere?

Step-by-Step Solution

Answer

The net charge on the sphere is approximately \(-7.49 \times 10^{-12} \mathrm{~C}\).

Step 1: Identify the Formula

To find the net charge on the sphere, we use Gauss's Law which states that the electric field times the area over a Gaussian surface equals the charge enclosed divided by the permittivity constant. The formula is: \[ E \cdot A = \frac{Q}{\varepsilon_0} \] where \(E\) is the electric field, \(A\) is the surface area of a sphere, \(Q\) is the charge, and \(\varepsilon_0 = 8.85 \times 10^{-12} \mathrm{~C}^2/\mathrm{N} \cdot \mathrm{m}^2\) is the permittivity of free space.

Step 2: Calculate the Surface Area

The electric field is given at \(15 \mathrm{~cm}\). We will use this as the radius of our Gaussian surface. The surface area \(A\) of a sphere is given by the formula \( A = 4\pi r^2 \). Substituting the radius, \( r = 0.15 \mathrm{~m}\): \[ A = 4\pi (0.15)^2 = 0.02827 \mathrm{~m}^2 \].

Step 3: Insert Known Values into Gauss's Law

Insert the values into Gauss's Law formula to solve for the charge \(Q\). We have \(E = 3.0 \times 10^3 \mathrm{~N}/\mathrm{C}\), \(A = 0.02827 \mathrm{~m}^2\), and \(\varepsilon_0 = 8.85 \times 10^{-12} \mathrm{~C}^2/\mathrm{N} \cdot \mathrm{m}^2\). Thus, \[ Q = E \cdot A \cdot \varepsilon_0 = (3.0 \times 10^3) \cdot (0.02827) \cdot (8.85 \times 10^{-12}) \].

Step 4: Calculate the Net Charge

Compute \( Q \) using the values from the previous step.\[ Q = (3.0 \times 10^3) \times 0.02827 \times 8.85 \times 10^{-12} \approx 7.49 \times 10^{-12} \mathrm{~C} \]. The charge is negative since the electric field is directed radially inward.

Key Concepts

electric field

The electric field is a vector field that surrounds electrically charged particles and exerts force on other charges in the field. Imagine it as an invisible force field around a charged object.
If you place another charged object within this field, it will experience an attractive or repulsive force. The direction of the electric field is the direction of the force that a positive test charge would feel.
In this particular exercise, the electric field is indicated as radial and directed inward, suggesting that the test charge would move towards the sphere. The magnitude of the electric field in our example is given as \(3.0 \times 10^{3} \text{ N/C}\). This tells us how strong the force will be felt per unit charge.

• Electric fields can be uniform or vary in space.
• Fields can be calculated using Gauss's Law, especially in symmetrical setups like spheres and cylinders.

permittivity of free space

Permittivity of free space is a crucial constant in electromagnetism, denoted by \(\varepsilon_0\). It quantifies how much electric field is "permitted" to penetrate through free space, influencing how electric charges interact. Mathematically, it appears in equations that describe electric fields, like Gauss's Law.

• Its value is \( 8.85 \times 10^{-12} \text{ C}^2/\text{N} \cdot \text{m}^2\).
• It acts as a proportionality constant that relates the electric field and electric flux to the charge enclosed by a Gaussian surface.

Understanding \(\varepsilon_0\) helps calculate how easily an electric field can influence a charge in space, significantly impacting the computation of forces in electric fields. In this exercise, \(\varepsilon_0\) is vital for finding the net charge via Gauss’s Law.

net charge calculation

Net charge calculation involves determining the total charge on an object based on electric field measurements. Using Gauss's Law is the most effective way in scenarios with high symmetry, such as spheres. Gauss's Law states:\[ E \cdot A = \frac{Q}{\varepsilon_0} \]Here, \(E\) is the electric field, \(A\) is the surface area of the Gaussian surface, \(Q\) is the net charge, and \(\varepsilon_0\) is the permittivity of free space.
In our case, the goal is to solve for \(Q\), the charge, knowing \(E\), \(A\), and \(\varepsilon_0\).
The steps are:

• Calculate the surface area \(A\) using the radius of the Gaussian surface, which in this problem is \(0.15 \text{ m}\).
• Insert the known values into the Gauss's Law equation.
• Perform the computation to find the charge \(Q\).

This process ultimately gives us the charge on the conducting sphere, in this case, \(-7.49 \times 10^{-12} \text{ C}\). The negative sign indicates the direction of the electric field is inward, signifying a negatively charged sphere.