Rotational Kinetic Energy

The rotational kinetic energy of an object with moment of inertia I is

K^{(rot)}=\frac{1}{2}I\omega^2,

where \omega is the angular velocity of the object.

Details

The rotational kinetic energy is just the sum of the kinetic energies for each little point mass in the object. It’s easy to see this, if we write the sum of each little mass m_i, moving with a velocity v_i a distance r_i away from the center of the rotating object:

K=\sum_i \frac{1}{2}m_iv_i^2=\sum_i \frac{1}{2}m_i (\omega r_i)^2=\frac{1}{2}\omega^2(\sum m_i r_i^2)=\frac{1}{2}I\omega^2

In this expression I’ve related the speed of each point with the speed of the whole body through v_i=\omega r_i. Then the last thing I did was use the definition of the moment of inertia, I=\sum m_i r_i^2.

References

Moore (2e): Unit C, pg 160

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Resistance

If there is a current I between two points which have a potential difference \Delta \phi, we define the resistance of the material through which the current is flowing as the ratio

R=\frac{\Delta \phi}{I}.

Details

This definition can be understood by thinking about what happens when an electric potential difference is present in a system. Because of the electrical interaction, charges will start to flow, from high potential to low potential if the charges are positive. But as soon as they start to flow, the microphysics of the material they are flowing through will govern how they move. Since we don’t know anything (at this level) about the behavior of the atoms of the material, we just look to see what the resulting current is and define the effective collective behavior of the atoms in the material as the resistance.

 

References

Wikipedia

Moore (3e): Unit E, pg 90

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Electrical Current

If electrical charge flows into or out of a region of space (or through an object), we say there is an electric current in that region. We can represent this mathematically in several different ways. If a small amount of charge dQ passes through some region in a time period dt, the current in that region is

I=\frac{dQ}{dt}.

Alternatively, if we have a charge density \rho in which the charges are moving at a velocity \vec{v}_d (called the drift velocity) through some region, we can specify the current density as

\vec{J}=\rho\vec{v}_d.

Details

Although the above expressions look pretty straight forward, it is actually a little difficult to be mathematically precise about these kinds of things without getting overly complicated. It is probably slightly better to speak of a “uniform charge passing through a surface A” in the case of the first expression, and a “uniform charge passing through a volume” for the second one. In that sense, the two expressions above are related via

\vec{I}=\vec{J}A.

References

Wikipedia

Moore (3e): Unit E, pg 84

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Conservation of Charge

In an isolated system we have a conservation of charge,

\Delta q=0.

Details

Just like conservation of energy, at this level we don’t need to know why this happens, just that it does. In this case “isolated system” literally means there are no interactions occurring that can change the net charge of the system as a whole. Also like energy conservation, this is part of a very important result called “Noether’s Theorem”, but that is well outside the scope of these notes.

References

Wikipedia

Moore (3e): Unit e, pg 9

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Electric Charge

The electric charge q is a fundamental property that objects can have. In some ways it is very much like mass – in an abstract sense we don’t care where the charge came from, but if we know it’s nonzero it will change the kinds of interactions we have in the system. A major difference between charge and mass is that mass can only be positive, but charge can be both positive and negative.

Details

Much can be said about the origin of the electric charge, but a real understanding requires some knowledge of microphysics (atomic, nuclear, or quantum). It’s true that we can experience the apparent creation of electric charge in our everyday lives (rubbing your feet on the carpet to build up a static charge, for instance), but unless someone at your local power plant is rubbing a giant carpet to power your television, something more detailed is going on.

References

Wikipedia

Moore (3e): Unit E, pg 4

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Electric Potential

The electric potential is defined as the electric potential energy V_e per unit charge:

\phi=\frac{V_e}{q}.

Details

The exact form of the expression above will be different depending on the distribution of the charge in the system. In general, one has to use the basic definition \vec{F}=-\nabla \phi to find the potential energy \phi before using the equation above. However, the electric potential a distance r from a point charge Q is

\phi=\frac{kQ}{r},

and many other distributions can be found by considering them as collections of small point charges dQ.

References

Wikipedia

Moore (3e): Unit E, pg 62

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Torque on an Electric Dipole

The torque on an object that has an electric dipole moment of \vec{p}_e due to an electric field \vec{E} is

\vec{\tau}_{dp}=\vec{p}_e\times \vec{E}.

Details

The source of this torque is easily seen to be due to the electric force on the two charges effectively separated by a distance  d:

\sum\vec{\tau}=\vec{r}_1\times \vec{F}_1+\vec{r}_2\times\vec{F}_2=\frac{1}{2}\vec{d}\times q\vec{E}+\frac{1}{2}(-\vec{d})\times (-q)\vec{E}=(q\vec{d})\times \vec{E},

and using the definition of the dipole moment \vec{p}_{dp}=q\vec{d}.

Reference

Wikipedia

Moore (3e): Unit E, pg 33

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