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Chapter 1: Problem 76

Hydrogen atom \(* *\) The neutral hydrogen atom in its normal state behaves, in some respects, like an electric charge distribution that consists of a point charge of magnitude \(e\) surrounded by a distribution of negative charge whose density is given by \(\rho(r)=-C e^{-2 r / a_{0}} .\) Here \(a_{0}\) is the Bohr radius, \(0.53 \cdot 10^{-10} \mathrm{~m}\), and \(C\) is a constant with the value required to make the total amount of negative charge exactly \(e\). What is the net electric charge inside a sphere of radius \(a_{0} ?\) What is the electric field strength at this distance from the nucleus?

Short Answer

Expert verified
The net electric charge within the sphere of radius \(a_{0}\) and the electric field strength at this distance from the nucleus can be determined by setting up integrals and applying the formulas concerning charge distributions and electric fields.

Step by step solution

01

Net Charge within the Sphere of Radius \(a_{0}\)

The total charge of a volume is the volume integral of its charge density. We can express the volume in spherical coordinates and determine the integral to find the total charge within the sphere of radius \(a_{0}\). The volume element in spherical coordinates is \( dV = r^2 sin(\theta)drd\theta d\phi \). The limits for r are [0, \(a_{0}\)], for \theta are [0, \(\pi\)], and for \phi are [0, 2\(\pi\)]. We setup the integral \(\int_{-\infty}^{+\infty} \rho(r) dV \) with these limits and solve.
02

Substituting for Charge Density and Volume Element

Substituting for \( \rho(r) \) and \( dV \) the integral becomes: \( \int \int \int_{V} (-C e^{-2 r / a_{0}}) * r^2 sin(\theta) dr d\theta d\phi \). And finally, we integrate over these limits to find the total charge within the sphere of radius \(a_{0}\).
03

Calculating Electric field at \(a_{0}\)

Given the symmetry of the problem, we can assume the electric field only has a radial component. The electric field \(E\) at a distance \(r\) caused by a volume charge density is given by the formula \( E = \frac{1}{4 \pi \epsilon_{0}}\frac{q_{enc}}{r^2} \). \(q_{enc}\) is the enclosed charge within the sphere of radius \(r\), which we found in the previous steps. Substituting these values we can calculate the magnitude of the electric field at \(a_{0}\).

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Key Concepts

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

Hydrogen Atom
When discussing a hydrogen atom, it is essential to remember that it is the simplest atom consisting of one proton and one electron. In its normal or ground state, the electron moves around the proton in an orbit that can be understood through quantum mechanics.
The hydrogen atom can be modeled as an electric charge distribution. This means that the positive charge exists at a point due to the proton, while the negative charge, thanks to the electron, is distributed around this point.
This distribution is not random. For a hydrogen atom in its normal state, the negative charge density follows the exponential function \(\rho(r) = -C e^{-2r/a_{0}}\). This representation helps us predict how the atom behaves in electric fields and how it interacts with other particles. Understanding the hydrogen atom's charged nature and distribution is foundational in fields like chemistry and spectroscopy.
Bohr Radius
The Bohr radius, denoted as \(a_{0}\), is a fundamental concept in the Bohr model of the atom. It is the average distance between the nucleus (proton) and the electron in a hydrogen atom in its ground state. Mathematically, the Bohr radius is approximately \(0.53 \times 10^{-10} \text{m}\).
This value is crucial for calculations involving charge distributions and electric fields within atoms. In the context of the exercise, the Bohr radius acts as a limit within which we calculate the net electric charge. It serves as a scaling factor, showing how electron clouds are distributed around a nucleus.
With the Bohr radius, scientists could make more precise calculations about atomic structures, electron behaviors, and predict chemical properties more accurately. It also allowed a deeper understanding of quantum mechanics since it provided a bridge between classical and quantum physics.
Electric Field Strength
Electric field strength is a measure of the force experienced per unit charge in the presence of an electric field. It is a vector quantity, often denoted by \(E\), and plays a critical role in understanding how charges interact.
For the hydrogen atom exercise, the electric field strength at a distance is determined using Gauss's law. This law relates the distribution of electric charge to the resulting electric field. In this scenario, the symmetry of the hydrogen atom's charge distribution means the field only has a radial component, simplifying calculations.
The expression for the electric field generated by a charge within a sphere at distance \(r\) is \(E = \frac{1}{4 \pi \epsilon_{0}}\frac{q_{enc}}{r^2}\), where \(q_{enc}\) represents the enclosed charge, and \(\epsilon_{0}\) is the permittivity of free space.
This relationship helps us understand the effects of the charge distribution around the nucleus, providing insights into how atoms interact with external electric fields.

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Most popular questions from this chapter

Stable equilibrium in electron jelly ** The task of Exercise \(1.77\) is to find the equilibrium positions of two protons located inside a sphere of electron jelly with total charge \(-2 e\). Show that the equilibria are stable. That is, show that a displacement in any direction will result in a force directed back toward the equilibrium position. (There is no need to know the exact locations of the equilibria, so you can solve this problem without solving Exercise \(1.77\) first.)

Field from a semicircle * A thin plastic rod bent into a semicircle of radius \(R\) has a charge \(Q\) distributed uniformly over its length. Find the electric field at the center of the semicircle.

Charges on a circular track Suppose three positively charged particles are constrained to move on a fixed circular track. If the charges were all equal, an equilibrium arrangement would obviously be a symmetrical one with the particles spaced \(120^{\circ}\) apart around the circle. Suppose that two of the charges are equal and the equilibrium arrangement is such that these two charges are \(90^{\circ}\) apart rather than \(120^{\circ} .\) What is the relative magnitude of the third charge?

Field in the end face Consider a half-infinite hollow cylindrical shell (that is, one that extends to infinity in one direction) with uniform surface charge density. Show that at all points in the circular end face, the electric field is parallel to the cylinder's axis. Hint: Use superposition, along with what you know about the field from an infinite (in both directions) hollow cylinder.

Maximum field from a ring ** A charge \(Q\) is distributed uniformly around a thin ring of radius \(b\) that lies in the \(x y\) plane with its center at the origin. Locate the point on the positive \(z\) axis where the electric field is strongest.
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