Physical Sciences 3
January 29, 2013
Welcome to Physical Sciences 3 •
What will we be learning this semester in PS3?
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Why are we learning all of this stuff?
http://cbs.fas.harvard.edu/science/connectome-project/brainbow
Physical Sciences 3
January 29, 2013
Gravitational Potential Energy •
Recall from PS2: a mass at a point y above the Earth’s surface has gravitational potential energy:
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Note that the value of the potential energy itself is not physically meaningful; only differences in potential energy can be measured:
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Why is the concept of gravitational potential energy useful? - Conservation of energy
- Relationship with kinetic energy
- Relationship with work
Physical Sciences 3
January 29, 2013
Force and Potential Energy •
In PS2, we derived a relationship between the potential energy of an interaction and the force associated with that interaction:
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For instance, we derived the spring force from the potential energy Uelastic = !kx2:
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Note that potential energy is a scalar, while force is a vector. In general, the force associated with an interaction is equal to (minus) the gradient of the potential energy:
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So, what is the force exerted by gravity on a mass m near the surface of the Earth?
Physical Sciences 3
January 29, 2013
The Gravitational Potential •
So, if there’s a mass present at some height y above the Earth’s surface, it has potential energy equal to mgy. But even if there’s no mass at that point, we can still define a gravitational potential !grav. What is the gravitational potential near the Earth?
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The gravitational potential is an important example of a scalar field. What does this mean? Can you give some other examples of scalar fields?
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One way to describe the gravitational force is: “The gravitational force on an object of mass m at some location is equal to its mass times (minus) the gradient of the gravitational potential at that location.” What does this mean? Can you show that this is true?
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Summary: The potential energy is equal to the mass times the gravitational potential. The force is equal to the mass times (minus) the gradient of the gravitational potential.
Physical Sciences 3
January 29, 2013
Electrical Potential and Potential Energy •
Just as there is a gravitational potential, there is also an electrical potential:
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What are the SI units of electrical potential?
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Is the actual value of the potential meaningful?
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What can you say about the electrical potential near the terminals of a 9-volt battery?
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What is the potential energy of an object with charge Q if the electrical potential is V?
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What are the SI units for charge? What about the SI units for energy? How are these units related?
Physical Sciences 3
January 29, 2013
Electrostatic Force •
We have a general relationship between potential energy and force:
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So, what is the force on a charge Q due to the electrical potential?
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We call this force the electrostatic force. The term “static” is included because the situation becomes more complicated if the potentials change a lot with time; we’re assuming here that the potentials don’t change (or change only slowly) over time.
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We can re-write the electrostatic force in terms of the gradient of the potential:
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What are the SI units of force? Can we make sense of all the units here?
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Summary: The potential energy is equal to the charge times the electrical potential. The force is equal to the charge times (minus) the gradient of the electrical potential.
Physical Sciences 3
January 29, 2013
Linear Potential Gradients •
The simplest kind of gradient is a linear gradient, where the potential changes linearly as a function of distance, starting at one value and ending at another:
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It’s easy to set up a linear potential gradient: 1. Take a battery or power supply with voltage "V. (Why is it a “delta”?) 2. Connect the terminals with a long, thin wire. (Careful! This is a “short circuit.”) 3. If the distance along the wire from one terminal to another is D, then the potential at any point along the wire is a linear function of x (where x = 0 at one end and x = D at the other). Find an expression for the potential, the potential gradient, and the force:
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Consider the following cases: A) I connect the terminals with a long piece of wire. B) I connect the terminals with a short piece of the same kind of wire. In which case will the force on the electrons in the wire be greater? Can we test it?
Physical Sciences 3
January 29, 2013
Examples •
Have you ever licked the terminals of a 9-volt battery?
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Why aren’t birds electrocuted when they sit on high-voltage wires?
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Electrical discharges in air: sparks and lightning
Physical Sciences 3
January 29, 2013
Summary •
Potential, Energy, and Force for Gravity (near the surface of the Earth): Gravitational potential (a field defined at all points in space):
!grav = gy Gravitational potential energy (of a mass m located at a particular point): Ugrav = m!grav = mgy Components of gravitational force (on a mass m located at a particular point):
Fgrav, y = !
Fgrav, x = !
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"U grav "y "U grav "x
= !m
= !m
"#grav "y "#grav "x
= !mg
=0
Potential, Energy, and Force for Electrostatics: Electrical potential (a field defined at all points in space): V(x, y, z) Potential at a point x between two terminals with potential difference "V and distance D (assuming that the material between the terminals is uniform): V (x) = !V
x D
Electrical potential energy (of a charge q located at a particular point): Uelec = QV Components of electrical force (on a charge q located at a particular point): Felec, x = !
"U elec "V = !Q "x "x
Force on a charge between two terminals with potential difference "V and distance D: Felec, y = !q
"V #V = !Q "x D
