Estimate the magnetic flux through the south- and west...

Estimate the magnetic flux through the south- and west facing windows of a house in British Columbia, where Earth's $\vec{B}$ field has a magnitude of $5.8 \times 10^{-5} \mathrm{T}$ and points roughly north with a downward inclination of $72^{\circ} .$ Explain how you made the estimates.

Estimate the magnetic flux through the south- and west facing windows of a house in British Columbia, where Earth's $\vec{B}$ field has a magnitude of $5.8 \times 10^{-5} \mathrm{T}$ and points roughly
north with a downward inclination of $72^{\circ} .$ Explain how you made the estimates.

Key Concepts

Magnetic Flux

Magnetic flux quantifies the amount of magnetic field passing through a given area. It is calculated as the dot product of the magnetic field vector and the area vector, which involves both the magnitude of the field and the cosine of the angle between the field and the normal to the surface.

Earth's Magnetic Field

Earth's magnetic field is a vector field that has both magnitude and direction. Its magnitude in this context is given as 5.8 × 10?? Tesla, and it is oriented approximately toward the north with a significant downward inclination, which affects how the field interacts with surfaces depending on their orientation.

Inclination and Projection

Inclination describes the angle between the Earth's magnetic field and the horizontal plane. A downward inclination of 72° indicates that the field makes a steep angle with the horizontal, meaning that when calculating flux through a vertical surface, one must project the field vector onto the normal of that surface using cosine of the relevant angle.

Dot Product in Flux Calculation

The dot product is a mathematical tool used to determine the component of one vector along the direction of another. In the context of magnetic flux, it is used to calculate how much of the Earth's magnetic field passes perpendicularly through a surface, taking into account both the magnitude of the field and the orientation of the window relative to the field direction.

Transcript

00:01 In this question, we are being asked to estimate the magnetic flux through a south and a west -facing window where the earth's magnetic field is going to have a magnitude of 5 .8 times 10 to the negative 5th tesla, and its direction is roughly north with a downward inclination of 72 degrees.

00:20 So let's first define our direction for this question.

00:24 So i'm going to define north as left and south as right, as you can see, in the legend here.

00:31 And i'm going to explain what it means to have a downward inclination of 72 degrees.

00:36 So say if you have a compass right here where the needle is pointing north, if you are to have a downward inclination, that means the needle is going to point downwards.

00:51 It is still pointing north, but the angle as it corresponds to the horizontal is now 72 degrees downward.

01:03 So that's how we're going to understand the direction of the magnetic field in this question.

01:10 And we're also going to roughly estimate the standard size of a window.

01:16 So i'm going to estimate it as a square with about 90 centimeters as its length.

01:23 And if i convert that to meters and then find the area of this window, then i get 0 .81 meter square.

01:30 As the area.

01:32 And since we're estimating flux, let's remind ourselves that the equation for magnetic flux is b times a times the cosine of the angle between these two vectors.

01:44 So what does that really mean? let me go ahead and redraw the picture with the sideways perspective.

01:52 So if this is my window right here, and if i have a magnetic field that is coming in at a, point that has a downward inclination, then i would say the magnetic field is coming down this way.

02:09 And what does area vector mean? so if this is the north facing window, then i'm going to say that it has an area vector that points north, and area vectors are always perpendicular to the surface of the area.

02:27 And that's why sometimes people call it the normal vector...

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