INT CALC Two positive point charges q are located on the y-axis a...

Question

INT CALC Two positive point charges q are located on the y-axis at y = ±a.e. Your answer to part d shows that an electron experiences a linear restoring force, so it will undergo simple harmonic motion. What is the oscillation frequency in GHz for an electron moving between two 1.0 nC charges separated by 2.0 mm?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Consider a situation where a proton is oscillating between two point charges each with a positive charge of 2.3 nano coons. These charges positioned 0.8 centimeters apart, exert forces on the proton that induce a simple harmonic motion. Calculate the frequency of this oscillator in megahertz hint the effective spring constant is two multiplied by K multiplied by Q multiplied by E all divided by D to the power of three where Q is the charge on one of the point charges E is the charge of the proton and D is the separation between the charges. Awesome. So it appears that we're asked to sol for the frequency of this oscillation in units of megahertz. And this is our final answer that we're ultimately trying to solve for. We're trying to figure out what the frequency of this oscillation is and we need to make sure it's in units of megahertz. Awesome. We're also given some multiple choice answers. And as we mentioned before, all of them are in units of megahertz. So let's read them off to see what our final answer might be. A is 16, B is 14 C, is 24 and D is 36. Awesome. So first off, let us note that this problem can be solved by understanding that the proton is oscillating due to the electrostatic force of the two point charges which will act as the restoring force in the simple harmonic motion. So we need to recall that in simple harmonic motion, the oscillation frequency can be written as F equals one divided by two multiplied by pi multiplied by the square root of K divided by M. And let's call this equation one. So equation one is the simple harmonic motion. The os of the simple harmonic oscillator. So this is our simple harmonic oscillator frequency equation to be specific. And let us also note that K equals two multiplied by KC multiplied by Q multiplied by E all divided by D three or D to the power of three I should say. So K in this case is the effective spring constant and KC is the charge or sorry not KC is Coolum constant. And Q, as we are told in the problem itself is the charge of one of the point charges and E is the charge of the proton. And then DD in this case is the separation between the charges and it's to the power of three awesome moving right along. Therefore, we can write equation one as F equals one divided by two, multiplied by pi multiplied by the square root of two multiplied by KC. Where KC once again is Kums constant multiplied by Q multiplied by E all divided by M multiplied by D to the power of three. And let's make a little note here. So we know that KC is Coolum constant but its numerical value is 8.99 multiplied by 10 to the power of nine. And its units are newtons multiplied by meters squared per Coolum square. And Q is the point charge and Q is equal to 2.3 nano Coolum and E is the elementary charge of the proton. And E the numerical value for that is 1.6 multiplied by 10 to the power of negative 19 coulombs. And D is the separation between the charges which is given to us as 0.8 centimeters, which we can convert this to meters and it will be equal to 0.008 m if we convert this to meters and you can use dimensional analysis or you can quickly look it up, but I just gave it to you. Awesome. And then we have M which M in this case is the mass of a proton, which numerically that value is 1.6 multiplied by 1.67. Let's be specific. So the mass of a proton is 1.67 multiplied by 10 to the power of negative 27 and its units are kilograms. Awesome. So now we can plug in all of our known variables and sulfur the frequency which is our final answer. So F is equal to, so F is equal to one divided by two, multiplied by pi multiplied by the square root of two multiplied by cools constant, which is 8.99 times 10 to the power of nine in its units are newtons multiplied by meters squared per Coolum squared, multiplied by 2.3 multiplied by 10 to the power of negative nine Coombs multiplied by 1.6 multiplied by 10 to the power of negative 19 cool Ls all divided by 1.67 multiplied by 10 to the power of negative 27 kg multiplied by 0.008. And we have to use the meters units value for D A set of centimeters to make sure all of our units cancel giving us our proper units that we're trying to solve for. So and don't forget that RD is to the power of three. So moving right along when we plug that into our calculator, we will find that F is equal to 14000 568.43. But this is in units of Hertz, but we need to convert Hertz to megahertz. So to do that, we need to m we need to divide so we need to divide our current value, which is in Hertz by 10 to the power of six. And when we do that, we will get 14, when we round to the nearest whole number 14 megahertz, which is our final answer. Hooray, we did it and this corresponds with multiple choice answer letter B 14 megahertz. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.

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