/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 27 A particle with a charge of \(-6... [FREE SOLUTION] | 91影视

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Chapter 24: Problem 27

A particle with a charge of \(-60.0 \mathrm{nC}\) is placed at the center of a nonconducting spherical shell of inner radius \(20.0 \mathrm{cm}\) and outer radius \(25.0 \mathrm{cm} .\) The spherical shell carries charge with a uniform density of \(-1.33 \mu \mathrm{C} / \mathrm{m}^{3}.\) A proton moves in a circular orbit just outside the spherical shell. Calculate the speed of the proton.

Short Answer

Expert verified
The speed of the proton moving in a circular orbit just outside the spherical shell can be computed by following the outlined steps.

Step by step solution

01

Calculate the total charge of the shell

Given the charge density \(\rho = -1.33 \times 10^{-6} \, \text{C/m}^3\), and the volume of the spherical shell can be calculated as \(\frac{4}{3}\pi(R_o^3 - R_i^3)\), where \(R_o\) is the outer radius and \(R_i\) is the inner radius, the total charge \(Q\) on the shell can be computed as \[Q = \rho \times \text{Volume of the shell} = \rho \times \frac{4}{3} \pi(R_o^3 - R_i^3). \]
02

Calculate the Electric Field Outside the Shell

The electric field \(E\) outside a spherical shell of radius \(r\) carrying charge \(Q\) is given by the formula \[E = \frac{k |Q|}{r^2},\] where \(k = 8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2\) is Coulomb's constant and \(r\) is the distance from the center of the sphere to the point in space where the electric field is being calculated (which in this case is the radius of the outer sphere, \(R_o\)).
03

Calculate the Centripetal Force

The only force acting on the proton is the force of electrostatic attraction (since it's moving in a circular orbit), which must balance the centripetal force necessary to keep the proton moving in its circular path. So, we set up the equation \[F_c = F_e,\] where \(F_c\) is the centripetal force and \(F_e\) is the electrostatic force. The centripetal force can be calculated using the formula \[F_c = m_p v^2 / R_o,\] where \(m_p\) is the mass of the proton and \(v\) is the speed of the proton (which we're trying to find). The electrostatic force can be calculated using the formula \[F_e = E \cdot |q|,\] where \(E\) is the electric field and \(|q|\) is the charge of the proton.
04

Solve for the Speed of the Proton

By substituting \(F_c = F_e\) for the centripetal force and the electrostatic force equations, we get the equation \[m_p v^2 / R_o = E \cdot |q|.\] We can solve this for \(v\) (the speed of the proton), which gives us \[v = \sqrt{(E \cdot |q| \cdot R_o) / m_p}.\] Substituting our calculated values for \(E\), \(|q|\), \(R_o\), and the known value for \(m_p\), we can find the numerical answer for \(v\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electrostatic Force
Electrostatic force is a fundamental concept of electromagnetism. It describes the force between two charged particles. According to Coulomb's Law, the electrostatic force (F_e) between two charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. The formula is given by: \[F_e = k \frac{|q_1 q_2|}{r^2}\]where q_1 and q_2 are the charges, r is the distance between them, and k is Coulomb's constant (8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2).

In the context of this exercise, a charged particle is placed inside a spherical shell, while a proton orbits outside it. The electrostatic force exerted by the charged shell on the proton depends on the charge of the shell and the proton, as well as the radius of the shell. This force is crucial for the proton鈥檚 orbit because it provides the necessary centripetal force to keep the proton moving in its circular path.
Centripetal Force
Centripetal force is the force required to make an object move in a circular path. It's always directed towards the center around which the object is moving. The formula for centripetal force (F_c) is:\[F_c = \frac{m v^2}{r}\]where m is the mass of the object, v is the velocity, and r is the radius of the circular path.

For a proton moving in a circular orbit outside the charged spherical shell, the centripetal force is provided by the electrostatic force. Hence, the balance between the electrostatic and centripetal force conditions allows for solving the speed of the proton. In this exercise, the formula is rearranged and combined with electrostatic principles to find the proton's speed.
Charge Density
Charge density (蟻) is a measure of the distribution of electric charge in a given volume. It鈥檚 represented by the formula:\[\rho = \frac{Q}{V}\]where Q is the total charge and V is the volume of the region. The charge density allows for calculating the total charge contained within a shell when the volume is known.

For a spherical shell, the volume V is the difference between the volumes of the outer and inner spheres:\[V = \frac{4}{3} \pi (R_o^3 - R_i^3)\]where R_o is the outer radius and R_i is the inner radius. Multiplying this volume by the charge density yields the total charge on the shell, crucial for assessing the electric field and related forces on particles surrounding the shell.

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Most popular questions from this chapter

An infinitely long insulating cylinder of radius \(R\) has a volume charge density that varies with the radius as $$\rho=\rho_{0}\left(a-\frac{r}{b}\right)$$ where \(\rho_{0}, a,\) and \(b\) are positive constants and \(r\) is the distance from the axis of the cylinder. Use Gauss's law to determine the magnitude of the electric field at radial distances (a) \(rR.\)

The electric field everywhere on the surface of a thin spherical shell of radius \(0.750 \mathrm{m}\) is measured to be \(890 \mathrm{N} / \mathrm{C}\) and points radially toward the center of the sphere. (a) What is the net charge within the sphere's surface? (b) What can you conclude about the nature and distribution of the charge inside the spherical shell?

(a) Using the mathematical similarity between Coulomb's law and Newton's law of universal gravitation, show that Gauss's law for gravitation can be written as $$\oint \mathbf{g} \cdot d \mathbf{A}=-4 \pi G m_{\mathrm{in}}$$ where \(m_{\text {in }}\) is the net mass inside the gaussian surface and \(\mathbf{g}=\mathbf{F}_{g} / m\) represents the gravitational field at any point on the gaussian surface. (b) Determine the gravitational field at a distance \(r\) from the center of the Earth where \(r

On a clear, sunny day, a vertical electric field of about \(130 \mathrm{N} / \mathrm{C}\) points down over flat ground. What is the surface charge density on the ground for these conditions?

A long, straight wire is surrounded by a hollow metal cylinder whose axis coincides with that of the wire. The wire has a charge per unit length of \(\lambda,\) and the cylinder has a net charge per unit length of \(2 \lambda\). From this information, use Gauss's law to find (a) the charge per unit length on the inner and outer surfaces of the cylinder and (b) the electric field outside the cylinder, a distance \(r\) from the axis.
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