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Chapter 21: Problem 46

Positive charge is spread uniformly over the surface of a spherical balloon \(70 \mathrm{cm}\) in radius, resulting in an electric field of \(26 \mathrm{kN} / \mathrm{C}\) at the balloon's surface. Find the field strength (a) \(50 \mathrm{cm}\) from the balloon's center and (b) \(190 \mathrm{cm}\) from the center. (c) What's the net charge on the balloon?

Short Answer

Expert verified
The net charge on the balloon can be calculated using Gauss's law. The electric field strength at 50cm and 190cm from the center of the balloon can also be calculated using the derived net charge and Gauss's law.

Step by step solution

01

Determine Total Charge on the Sphere

We can use Gauss's Law to find the total charge on the sphere. The electric field at the surface of the sphere is given as \(E_{surface}=26 kN/C\), and the radius is \( r=70 cm = 0.7 m\). The formula for Gauss's Law is \( \Phi_E = E \cdot A = q/ \epsilon_0 \), where \(q\) represents the total charge and \(A\) represents the area. After rearranging this, we get the charge as \(q = E_{surface} \cdot 4 \pi r^2 \cdot \epsilon_0 \), with \(\epsilon_0=8.85 \times 10^{-12} C^2/(N m^2)\)
02

Calculate the Field Strength 50cm from the Center

According to Gauss's Law, inside a sphere, the electric field strength is \( E_1 = q / (4 \pi \epsilon_0 r_1^2) \), where \( r_1 = 50 cm = 0.5 m\). Substitute the total charge calculated from Step 1 and the given value of \( r_1 \) to get the electric field strength 50cm from the center of the sphere.
03

Calculate the Field Strength 190cm from the Center

Similarly, we can use Gauss's law to calculate the electric field strength at 190cm from the center of the sphere, using the equation \( E_2 = q / (4 \pi \epsilon_0 r_2^2) \), where \( r_2 = 190 cm = 1.9 m\). Substitute the total charge calculated from Step 1 and the given value of \( r_2 \) to get the electric field strength 190cm from the center.

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Most popular questions from this chapter

Why must the electric field be zero inside a conductor in electrostatic equilibrium?

The charge density within a uniformly charged sphere of radius \(R\) is \(\rho=\rho_{0}-a r^{2},\) where \(\rho_{0}\) and \(a\) are constants and \(r\) is the distance from the center. Find an expression for \(a\) such that the electric field outside the sphere is zero.

What's the electric field strength in a region where the flux through a \(1.0 \mathrm{cm} \times 1.0 \mathrm{cm}\) flat surface is \(65 \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C},\) if the field is uniform and the surface is at right angles to the field?

A net charge of \(5.0 \mu \mathrm{C}\) is applied on one side of a solid metal sphere \(2.0 \mathrm{cm}\) in diameter. Once electrostatic equilibrium is reached, and assuming no other conductors or charges nearby, what are (a) the volume charge density inside the sphere and (b) the surface charge density on the sphere?

What's the approximate field strength \(1 \mathrm{cm}\) above a sheet of paper carrying uniform surface charge density \(\sigma=45 \mathrm{nC} / \mathrm{m}^{2} ?\)
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