and an elecron of charge -Q = A hydrogen atom Is made up of a proton of charge +Q = 1.6x10 1.6x10-19C. The proton may be regarded as a point charge at r = 0 the center of the atom. The motion of the elecron causes its charge to be "smeared out into a spherical distribution around the proton, so that the electron is equivalentto a charge per unitvolume of (r) = -(Q/na})e"lao, where a, = 5.29x10-11 m is called the Bohr radius. (a) Find the total amount of the hydrogen atom's charge that is endosed within a sphere with radius centered on the proton. (b) Find the elecricfield (magnitude and diredtion) caused by the charge of the hydrogen atom as a function of (c) Make a graph as a function of r of the ratio of the electric-field magnitude to the magnitude of the field due to the proton alone Set up. 1. 1. The charge distribution in this task is spherically symmetric, so you can solve it using Gauss's law. 2. The charge within a sphere of radius rindudes the proton charge Qplus the portion of the eledron charge distribution that lies within the sphere. The electron charge distribution is not uniform, so the charge endosed within a sphere of radius r is not simply the charge density multiplied by the volume of the sphere. Instead, you'll have to do an integral. 3. Consider a thin spherical shell centered on the proton, with radius r and infinitesimal thickness dr. Since the shell is so thin, every point within the shell is at essentially the same radius from the proton. Hence the amount of electron charge within this shell is equal to the eledtron charge density p(r) at this radius multiplied by the volume dV of the shell. What is dV in terms of r? 4. The total electron charge within a radiusrequals the integral of from to . Set up this integral (but don't solve it yet), and use it to write an expression for the total charge (induding the
and an elecron of charge -Q = A hydrogen atom Is made up of a proton of charge +Q = 1.6x10 1.6x10-19C. The proton may be regarded as a point charge at r = 0 the center of the atom. The motion of the elecron causes its charge to be "smeared out into a spherical distribution around the proton, so that the electron is equivalentto a charge per unitvolume of (r) = -(Q/na})e"lao, where a, = 5.29x10-11 m is called the Bohr radius. (a) Find the total amount of the hydrogen atom's charge that is endosed within a sphere with radius centered on the proton. (b) Find the elecricfield (magnitude and diredtion) caused by the charge of the hydrogen atom as a function of (c) Make a graph as a function of r of the ratio of the electric-field magnitude to the magnitude of the field due to the proton alone Set up. 1. 1. The charge distribution in this task is spherically symmetric, so you can solve it using Gauss's law. 2. The charge within a sphere of radius rindudes the proton charge Qplus the portion of the eledron charge distribution that lies within the sphere. The electron charge distribution is not uniform, so the charge endosed within a sphere of radius r is not simply the charge density multiplied by the volume of the sphere. Instead, you'll have to do an integral. 3. Consider a thin spherical shell centered on the proton, with radius r and infinitesimal thickness dr. Since the shell is so thin, every point within the shell is at essentially the same radius from the proton. Hence the amount of electron charge within this shell is equal to the eledtron charge density p(r) at this radius multiplied by the volume dV of the shell. What is dV in terms of r? 4. The total electron charge within a radiusrequals the integral of from to . Set up this integral (but don't solve it yet), and use it to write an expression for the total charge (induding the
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Transcribed Image Text:A hydrogen atom is made up of a proton of charge +Q = 1.6x10-19C and an eledron of charge -Q =
1.6x10-19C. The proton may be regarded as a point charge at r = 0 the center of the atom. The motion of the
eledtron causes its charge to be "smeared ouť into a spherical distribution around the proton, so that the electron is
equivalentto a charge per unitvolume of (7) =-(Q/ra?)e"lao, where a, = 5.29x10-11 m is called the
Bohr radius. (a) Find the total amount of the hydrogen atom's charge that is endosed within a sphere with radius
centered on
the proton. (b) Find the eledricfield (magnitude and diredion) caused by the charge of the hydrogen atom as a
fundtion of (c) Make a graph as a function of rof the ratio of the eledtricfield
magnitude to the magnitude of the field due to the proton alone
Set up:
1. 1. The charge distribution in this task is spherically symmetric, so you can solve it using
Gauss's law.
2. The charge within a sphere of radius r indudes the proton charge Qplus the portion of the
elecron charge distribution that lies within the sphere. The electron charge distribution is not
uniform, so the charge endosed within a sphere of radius r is not simply the charge density
multiplied by the volume of the sphere. Instead, you'll haveto do an integral.
3. Considera thin spherical shell centered on the proton, with radius r and infinitesimal thickness
dr. Since the shell is so thin, every pointwithin the shell is at essentially the same
radius from the protorn. Hence the amountof eledron charge within this shell is equal to the
eledtron charge density p(r) at this radius multiplied by the volume dV of the shell. What is dV
in terms of r?
4. The total elecron charge within a radiusrequals the integral of from to . Set up this integral
(but don't solve it yet), and use it to write an expression for the total charge (induding the
proton) within a sphere of radius r
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