You're driving at 95 km / h in a direction 35^∘ east of north, in a region where the earth's mag

step 1 For this problem, you are driving in a car at 95 kilometers per hour. This is going 35 degrees e

You're driving at 95 km / h in a direction 35^∘ east of...

You're driving at 95 $\mathrm{km} / \mathrm{h}$ in a direction $35^{\circ}$ east of north, in a region where the earth's magnetic field of $5.5 \times 10^{-5} \mathrm{T}$ is horizontal and points due north. If your car measures 1.5 $\mathrm{m}$ from its underbody to its roof, calculate the induced emf between roof and underbody. (You can assume the sides of the car are straight and vertical.) Is the roof of the car at a higher or lower potential than the underbody?

Key Concepts

Motional Electromotive Force (EMF)

Motional EMF is the voltage generated across a conductor when it moves through a magnetic field. This effect arises because the moving charges in the conductor experience a magnetic Lorentz force, which separates positive and negative charges, thereby creating an electric potential difference along the length of the conductor.

Lorentz Force

The Lorentz force is the force exerted on a charged particle moving in a magnetic field, given by the cross product of the particle's velocity and the magnetic field. This force is central to understanding how motion through a magnetic field can induce an electromotive force in a conductor.

Vector Cross Product and Right-Hand Rule

The vector cross product is used to determine both the magnitude and the direction of the force on the moving charges. The right-hand rule is a practical tool in applying the cross product, allowing one to find the direction of the induced force or emf, which is perpendicular to both the velocity of the moving conductor and the magnetic field.

Unit Conversion and Dimensional Consistency

Ensuring that units are consistent throughout any calculation is essential in physics. For instance, converting speeds from kilometers per hour to meters per second ensures that the resulting calculation for emf has the correct dimensions and magnitude.

Transcript

00:02 For this problem, you are driving in a car at 95 kilometers per hour.

00:07 This is going 35 degrees east of north.

00:11 The car is 1 .5 meters based roof, and the earth's magnetic field is pointing north with the value of 5 .5 times 10 to the negative 5 tesla.

00:20 Our goal is to find the induced emf between the roof and the underbody.

00:26 So let's label two points on here.

00:30 We'll do that top corner is point a.

00:33 And the bottom corner is point b.

00:37 Now we know if there is a constant electric field between a and b, then the potential difference, vab, would be given by whatever that electric field is, multiplied by the displacement between the two, which i'm denoting as h.

00:56 Now, is there an electric field between the two? well, to get an idea of this, let's take the perspective of looking, at the car from behind.

01:08 So that, draw our little rectangular body.

01:13 We've got our little tires down here.

01:19 And then let's consider, actually i'm going to blow this up a little bit because we're going to do some free body diagrams.

01:26 So let's consider what is happening to our protons and our electrons.

01:34 And i'll just write for emphasis, this is our back view.

01:36 So for this whole body moving through an electric field, we know that our magnetic field exerts a force on these protons and electrons given by whatever the charge is, q, the velocity, magnitude of the b field, and then the sign of the angle between that b field and our velocity.

02:06 So here, we're told that that velocity is 95 kilometers per hour.

02:16 And we're also told what that angle is.

02:18 It's 35 degrees.

02:21 So now all we have to do is use our right hand rule.

02:24 If you point your index finger along the velocity, your middle finger along the b field, then your thumb is pointing up.

02:32 And since that charge for the proton is positive, that up gives us our direction of force, for the proton.

02:40 And since this is the back view, the up is up...