An automobile with a radio antenna 1.0 meters long travels at 100.0 kilometers per hour in a location where the Earth’s horizontal magnetic field is 5.5 times 10 to the negative fifth tesla. What is the maximum possible emf induced in the antenna due to this motion?
We’ll call the length of the radio antenna, 1.0 meters, 𝑙. And we’ll call the car’s speed, 100.0 kilometers per hour, 𝑣. The horizontal magnetic field of 5.5 times 10 to the negative fifth tesla we’ll call 𝐵. We want to solve for the maximum possible emf induced in the antenna due to the motion of the car. We’ll call this emf 𝜀.
Let’s start with the sketch of our scenario. Our car with its antenna pointed straight up and down moves along with speed 𝑣 on the road. We’re told that at this location, there is a horizontally oriented magnetic field, 𝐵, that we’ve drawn going into the page. We want to solve for the emf that is induced in the car’s radio antenna due to its motion.
And to do that, we’ll work from Faraday’s law. This law tells us that the emf induced in a loop, 𝜀, is equal to the change in magnetic flux through that loop per unit time. And we further recall that magnetic flux is equal to magnetic field 𝐵 multiplied by the area through which it acts. Combining these equations, we can write that the induced emf is equal to the change in the product of 𝐵 times 𝐴 per unit time.
But we know from our problem scenario that 𝐵 does not change. It’s a constant all throughout. So we can rewrite our numerator as 𝐵 times 𝑑𝐴, the magnetic field times the change in area. But what is that change in area 𝑑𝐴? Well if we look up close at the motion of the antenna as it moves, we know that the antenna has a length 𝑙 and that in any moment of time, 𝑡, horizontally the antenna moves a distance 𝑣 times 𝑡.
If we create a box defined by these sides, then the area of that box, 𝐴, is equal to the area that we’ll use in our Faraday’s law equation. So we can again rewrite our numerator as 𝐵 the magnetic field times 𝑑𝑙𝑣𝑡. Looking at the terms and parentheses, we know that 𝑙, the length of the antenna, doesn’t change and neither does the speed of the car, 𝑣. Only 𝑡 changes, so we rewrite our numerator as 𝐵 times 𝑙 times 𝑣𝑑𝑡.
But now we see that we have a 𝑑𝑡 in both the numerator and denominator and can cancel them out. So we see now that our induced emf is equal to the magnetic field times the length of the antenna times the speed of the car, 𝑣. When we plug in for these values, we’re careful to rewrite the speed 𝑣 in units of meters per second. We do that by multiplying the given 𝑣 by this conversion factor of 1000 meters per 3600 seconds.
When we enter these values on our calculator, we find that, to two significant figures, the induced emf is 0.0015 volts. That’s the emf that can be induced in this car radio antenna due to the car’s motion.