The wave equation describes disturbances in a medium that propagate with a constant speed: or [2nd time derivative] - [constant] x [2nd spatial derivative] If the equation of motion for a system can be put into this form, then that system will allow waves that propagate with speed c: (speed] - V[constant] Use this idea to analyze the speed of waves in the systems that follow. Maxwell's Equations: When developing his mathematical description of electricity and magnetism, Maxwell discov- ered that electric and magnetic fields could travel through empty space. The equation of motion for the electric field in this situation is where 4g is the permeability of free space and co is the permittivity of free space. (a) Based on Maxwell's analysis, what is the speed at which the electric field propagates through empty space? (Report your result in terms of co and Ho, and then evaluate and report the numerical result.) Sound in Metals: In class, we found that applying Newtonian mechanics to series of masses on spring led to a wave equation: where m was the mass of a particle, k was the spring constant, a was the equilibrium length of a spring, and u(x,t) describes the displacement of the particle at location x from its equilibrium position at time t. This is a decent model for studying sound waves in metals. (b) The speed of sound in copper is 4.76 km/s, and the interatomic spacing in crystalline copper is about 0.360 nm. The molar mass of copper is 63.546 g/mol. Based on these data, what is the interatomic “spring constant" k between copper atoms?
The wave equation describes disturbances in a medium that propagate with a constant speed: or [2nd time derivative] - [constant] x [2nd spatial derivative] If the equation of motion for a system can be put into this form, then that system will allow waves that propagate with speed c: (speed] - V[constant] Use this idea to analyze the speed of waves in the systems that follow. Maxwell's Equations: When developing his mathematical description of electricity and magnetism, Maxwell discov- ered that electric and magnetic fields could travel through empty space. The equation of motion for the electric field in this situation is where 4g is the permeability of free space and co is the permittivity of free space. (a) Based on Maxwell's analysis, what is the speed at which the electric field propagates through empty space? (Report your result in terms of co and Ho, and then evaluate and report the numerical result.) Sound in Metals: In class, we found that applying Newtonian mechanics to series of masses on spring led to a wave equation: where m was the mass of a particle, k was the spring constant, a was the equilibrium length of a spring, and u(x,t) describes the displacement of the particle at location x from its equilibrium position at time t. This is a decent model for studying sound waves in metals. (b) The speed of sound in copper is 4.76 km/s, and the interatomic spacing in crystalline copper is about 0.360 nm. The molar mass of copper is 63.546 g/mol. Based on these data, what is the interatomic “spring constant" k between copper atoms?
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![1 Speed of Propagation
The wave equation describes disturbances in a medium that propagate with a constant speed:
or
[2nd time derivative] - [constant] x [2nd spatial derivative]
If the equation of motion for a system can be put into this form, then that system will allow waves that propagate with
speed c:
(speed) - V[constant]
Use this idea to analyze the speed of waves in the systems that follow.
Maxwell's Equations: When developing his mathematical description of electricity and magnetism, Maxwell discov-
ered that electric and magnetic fields could travel through empty space. The equation of motion for the electric field in
this situation is
where 4, is the permeability of free space and 6o is the permittivity of free space.
(a) Based on Maxwell's analysis, what is the speed at which the electric field propagates through empty space? (Report
your result in terms of eo and Ho, and then evaluate and report the numerical result.)
Sound in Metals: In class, we found that applying Newtonian mechanics to series of masses on spring led to a wave
equation:
każu
əx
a²u
where m was the mass of a particle, k was the spring constant, a was the equilibrium length of a spring, and u(x,t)
describes the displacement of the particle at location x from its equilibrium position at time t. This is a decent model for
studying sound waves in metals.
(b) The speed of sound in copper is 4.76 km/s, and the interatomic spacing in crystalline copper is about 0.360 nm. The
molar mass of copper is 63.546 g/mol. Based on these data, what is the interatomic “spring constant" k between
copper atoms?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5c23085c-d041-4baf-a967-24cb96374560%2Fe8201563-2d22-4428-ae81-f2f21adfbc29%2Fqlcmdh_processed.png&w=3840&q=75)
Transcribed Image Text:1 Speed of Propagation
The wave equation describes disturbances in a medium that propagate with a constant speed:
or
[2nd time derivative] - [constant] x [2nd spatial derivative]
If the equation of motion for a system can be put into this form, then that system will allow waves that propagate with
speed c:
(speed) - V[constant]
Use this idea to analyze the speed of waves in the systems that follow.
Maxwell's Equations: When developing his mathematical description of electricity and magnetism, Maxwell discov-
ered that electric and magnetic fields could travel through empty space. The equation of motion for the electric field in
this situation is
where 4, is the permeability of free space and 6o is the permittivity of free space.
(a) Based on Maxwell's analysis, what is the speed at which the electric field propagates through empty space? (Report
your result in terms of eo and Ho, and then evaluate and report the numerical result.)
Sound in Metals: In class, we found that applying Newtonian mechanics to series of masses on spring led to a wave
equation:
każu
əx
a²u
where m was the mass of a particle, k was the spring constant, a was the equilibrium length of a spring, and u(x,t)
describes the displacement of the particle at location x from its equilibrium position at time t. This is a decent model for
studying sound waves in metals.
(b) The speed of sound in copper is 4.76 km/s, and the interatomic spacing in crystalline copper is about 0.360 nm. The
molar mass of copper is 63.546 g/mol. Based on these data, what is the interatomic “spring constant" k between
copper atoms?
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