An electric dipole consists of 1.0 g spheres charged to ±2.0 n...

Question

An electric dipole consists of 1.0 g spheres charged to ±2.0 nC at the ends of a 10-cm-long massless rod. The dipole rotates on a frictionless pivot at its center. The dipole is held perpendicular to a uniform electric field with field strength 1000 V/m, then released. What is the dipole's angular velocity at the instant it is aligned with the electric field?

Accepted Answer

Welcome back. Everyone in this problem. A science museum exhibit has a 30 centimeter long road with 5 g metal spheres charged to positive and negative 10 nano colons. At the end, the rod is mounted on a frictionless pivot and is released in a uniform 500 volts per meter electric field. We want to calculate the rod's angular velocity when it aligns with the electric field. For our answer choices A is 0.22 radiance per second. B is 0.12 C is 0.15 and D is 0.32. And all of these are in units for radiance per second. Now let's try to visualize what's really going on here. So we have two metal spheres, OK? Two metal spheres that are attached to the end of a 30 centimeter long rod. So we can represent it using this diagram here. OK. We know that for a rod, its mass is 5 g. So let's make a note of all the information we have. Its mass is 5 g which in si units that's five times 10 to the negative 3 kg. We know that the distance between the charges positive and negative is 30 centimeters, 30 times 10 to the negative 2 m. And we also know that each sphere is charged to 10 nano coons at the ends. So we could say on this end, it's negative 10 nano coulombs and on the other end, on the right, it's positive 10 nano coulombs doesn't really matter because what we're really interested in is the magnitude of our charge. What else do we know? Well, we also know that it's in an electric field in a uniform electric field that's 500 volts per meter. So the strength of the electric field is 500 volts per meter. And we could visualize it here in these red lines. OK, as our electric field, and we want to calculate the rod's angular velocity when it aligns with the electric field. In other words, our rod is going to spin until it aligns with our electric field here that we can imagine our rod is probably gonna look something like this when it's finished. So what will be the angular velocity at that point? That's the question we're really asking. So let's ask ourselves, what do we know about a spinning rod and angular velocity? Well, we know that the kinetic energy in a rod depends on the angular velocity, how fast it's spinning. So recall that are kinetic energy is going to be equal to a half of I multiplied by omega squared where I represents the moment of inertia of the dipole and omega represents its angular velocity. Therefore, that means if we want its angular velocity, then we can multiply its kinetic energy by two. OK. We can multiply kinetic energy by two divided by the moment of inertia of the dipole because in essence, this is a dipole two opposite charges at either end. And then we can find the square root of that result. No, the thing is that's great And all, but we don't know what the kinetic energy is and we don't know what the moment of inertia of the dipole is. So we need to figure both of those out. So let me put the kinetic energy in blue and let me shade the moment of inertia of the dipole red. And let's see if we can figure out both of those in order to solve for the angular velocity. Let's start with our kinetic energy. What else do we know about kinetic energy in this scenario? Well, based on the law of energy con conservation, we know that the energy that we start out with must be equal to the energy that we end with. We're talking about a spinning rod here. So our energy involved would have been its kinetic energy, of course, its potential energy. And since we're assuming the rod is, since the rod is mounted on a frictionless pivot, we're assuming that no energy is lost the heat or anything else. So solving for, we want to get or getting an expression. OK. For our kinetic energy using the law of energy conservation e energy conservation that tells us then that as we said, the energy we ended with must must be equal to the energy we start with. So our initial kinetic energy and initial and initial potential energy or rather sorry, the energy we ended with. So our final kinetic energy forgive me and our final potential energy must be equal to our initial kinetic energy and our initial potential energy. Now off the bat, do we know anything about these energies here? Well, we know that because the rod is initially stationary and finally aligns with the electric field, the initial kinetic energy would be equal to zero. OK. So our initial kinetic energy equals zero, we also know that a dipole potential energy U is calculated by negative pe multiplied by the cosine of theta where P is the dipole moment E is the electric field and theta is the angle between the dipole and the electric field. We also know that that dipole moment. So we're making a few notes here, we also know that the dipole moment P will be equal to or it's the product of the charge and the separation between the charges we'll get back to that. But we're just making a note of that information here. So since we know that that means then we can rewrite our formula to say that our kinetic energy will be equal to the difference between our initial potential energy and our final potential energy. Using what we know about our potential electric or rather using what we know about our di po potential energy. OK. Since U equals negative P multiplied by the cosine of theta, then we can think about what will happen in the for the initial potential energy and the final potential energy. So let's start with our initial potential energy initially notice that the rod is perpendicular to the field. OK. So that means the would be 90 degrees. So in that case, our initial potential energy would be equal to negative pe multiplied by the cosine of 90 degrees and the cosine of 90 degrees equals zero. So our initial potential energy equals zero. On the other hand, our final potential energy remember at this point, the rod is not aligned with the electric field. So theta equals 90 degrees. So that means our final potential energy will be negative pe multiplied by the cosine of zero degrees and the cosine of zero degrees equals one. So our final potential energy equals negative pe, we now have expressions for both of these. So we can put them into our formula for the kinetic energy because now that tells us our kinetic energy will be equal to zero minus negative pe which that will be equal to positive pe. Finally recall that p that is our dipole moment is equal to our charge multiplied by its separation. So actually, we can rewrite this as Q multiplied by s multiplied by E because remember both of the charges are at the end of the rod and the rod is 30 centimeters which we know now we could go ahead and solve for our kinetic energy. We know all of these. But let's hold on to that for a bit. And let's see if we can figure out our dipole moment of inertia. And then we're going to combine all the expressions into one formula to simplify. So let's put this into our back pocket for a bit. So we know how the kinetic energy next, we need to find the moment of inertia of the dipole. So what do we know about that? Well, so solving for, I recall that the moment of inertia of the dipole can be calculated using the formula to M multiplied by half of S squared where M is the mass of the sphere and S is the length of the rod. Now, if we expand this expression, we'll find that we get two M multiplied by a half of S squared, which would be S squared divided by four. And if we simplify it even even further, we'll end up with S squared. OK, we'll end up with M multiplied by S squared divided by two. Now, we have an expression for the moment of inertia of the dipole. Since we have expressions with known values for both of these terms. Are both of these variables. Now, we can go ahead and solve for the angular velocity. This is the big moment. So let's use everything that we have. So this now tells us then that our angular velocity is going to be equal to the square root of our kinetic energy, which is going to be are two multiplied by our kinetic energy. So that's gonna be two multiplied by Q multiplied by S multiplied by E and all of that will be divided by the moment of inertia of our dipole, which is a half of MS squared. No, we can do some canceling here because two divided by a half equals four S into S squared goes SS times. So if we go ahead, then we can write this even further to have our angular velocity as the square root of four multiplied by Q multiplied by E all divided by M multiplied by S now that we have all our known values we can go ahead and sell. So now this tells us then that our angular velocity is going to be equal to the square root of four multiplied by our charge. Recall that our charge was 10 nano colons. OK. Multiplied by our electric field which is 500 volts per meter. And all of that is going to be divided by our mass which is 5 g, five times 10 to the negative 3 g multiplied by our mass. Sorry, our distance between the rod s which is 30 times 10 to the negative 2 m. And then we're going to find the square root of all of that expression. So let's go ahead and put that into our calculators. When you do that. And when you round it to two decimal places, you'll find that the angular velocity equals approximately 0.12 radiance per second. So in other words, the angular velocity of the rod when it aligns with the electric field is approximately 0.12 radiance per second. When we look back on our answer choices that would have been answer choice B thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Electric Dipole Moment

Torque in an Electric Field

Conservation of Energy

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