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This video is about charges in orbit.

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We learned in subtopic 7.3 that in the rather forward model of the atom, negatively charged

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electrons orbit the positively charged nucleus in circular orbits.

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Even though this model is not fully correct, which we will discuss in more detail in subtopic

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12.1, for now we'll work with this model and assume that the electron remains in a

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circular orbit due to the force between the positively charged nucleus and the negatively

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charged electron.

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Let's examine the simplest atom, which is the hydrogen atom.

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In the hydrogen atom, a single electron orbits a proton.

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Let's assume that the charge of the electron is negative lowercase q, the charge of the

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proton is positive uppercase q, the distance between the electron and the proton, so the

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orbital radius is r and that the mass of the electron is m.

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In subtopic 5.1 we learned about Coulomb's law.

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In this subtopic we returned to this law and once again saw the equation that expresses

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this law.

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In subtopic 6.1 we learned about circular motion.

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We discovered that when an object moves in a circle at constant speed, a centripetal

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force is present.

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Here is the equation to calculate the centripetal force.

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In our example here, the centripetal force is provided by the force that the proton exerts

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on the electron.

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Therefore we can write that mb squared over r is equal to k uppercase q times lowercase

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q divided by r squared.

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You might have noticed that I omitted the negative sign that would appear on the right

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hand side of this equation when we substitute negative q for the charge of the electron.

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The negative sign here shows that the force between the electron and the proton is attractive.

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Since for this calculation we are interested in the magnitude of the force, I omitted the

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negative sign from the equation.

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Moving further we can simplify by r and after dividing by m and taking the square root we

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get that v is equal to k times uppercase q times lowercase q divided by mr under the

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square root.

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So this is the expression for the orbital speed when a charge orbits another charge.

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In our example here we can take this one step further because the magnitude of the electrons

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and the protons charge is the same.

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It is the elementary charge.

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This means that we can replace both uppercase q and lowercase q by e, the symbol of the

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elementary charge.

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This gives us that v is equal to k times e squared over mr under the square root.

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Let's move on to calculate the total energy of the orbiting electron.

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Et, the total energy of the electron, has two components, potential energy and kinetic

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energy.

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Here is the equation that we learned for electrical potential energy earlier in this subtopic

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and here is the kinetic energy equation from subtopic 2.3.

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Substituting we get that Et is equal to k times positive q times negative q divided

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by r.

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Since here potential energy is negative, I will not omit the signs of the charges when

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calculating the potential energy.

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To this we will add the kinetic energy which is 1 half times m times v squared, where v

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is the orbital speed of the electron that we just found.

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So we get k times uppercase q times lowercase q divided by mr under the square root to the

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power of 2.

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Working further we have that Et is equal to negative k uppercase q lowercase q over r

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plus 1 half and in the expression for the kinetic energy the square root and the square

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cancel and we can also cancel m giving us 1 half times k times uppercase q times lowercase

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q divided by r.

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Combining the two terms we get that Et is equal to negative k times uppercase q times

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lowercase q over 2r, which again considering that the charge of the electron and the proton

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is equal to the elementary charge, we get that the total energy is equal to negative

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k times e squared over 2r.

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Note that both the orbital speed and the total energy are very similar to what we found

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for the orbital speed and the total energy for a satellite that is in orbit around the

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planet.

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This once again shows that fields in physics have very similar characteristics.

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This wraps up our discussion about charges in orbit and completes the final video of

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Subtopic 10.2.

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In the first video of Subtopic 11.1 we'll start learning about electromagnetic induction.