A +6.00 μC point charge is moving at a constant 8.00 ×10^6 m / s in the +y -direction, relative

A +6.00 μC point charge is moving at a constant 8.00 ×10^6...

Key Concepts

Spatial Dependence of Magnetic Fields

The magnitude of the magnetic field from a moving charge decreases with the square of the distance from the charge, as indicated by the r² term in the denominator of the formula. This spatial dependence means that the field strength varies significantly with the position of the observation point relative to the charge. Understanding this relationship is important for analyzing how the magnetic field behaves in different regions of space around a moving charge.

Vector Cross Product and Right-Hand Rule

The vector cross product is central to determining the direction of the magnetic field generated by a moving charge. The cross product between the velocity vector and the position unit vector (r?) gives a vector whose direction is orthogonal to both. The right-hand rule is used to establish this direction, with the thumb pointing in the direction of the velocity and the fingers pointing towards the field point, the resulting magnetic field vector then emerges from the palm. This geometric interpretation is vital in visualizing and calculating the field’s orientation.

Magnetic Field of a Moving Charge

A moving point charge generates a magnetic field as a result of its motion, which can be calculated using a formula derived from the Biot–Savart law. The magnetic field is expressed as B = (??/4?) * (q v × r?)/r², where q is the charge, v is its velocity, r? is the unit vector from the charge to the field point, and r is the distance between them. This concept highlights how the dynamic nature of charge movement results in a magnetic influence in the surrounding space.

Biot–Savart Law

The Biot–Savart law is a fundamental principle in electromagnetism that relates currents (or moving charges) to the magnetic fields they produce. It states that the contribution to the magnetic field at a given point is proportional to the current element and inversely proportional to the square of the distance from the element, with the direction given by the cross product of the current element with the unit vector pointing from the element to the point of observation. This law underpins the calculation of magnetic fields for various current distributions.

Transcript

0:00 All right, everyone.

00:01 So we're going to be determining the direction and magnitude of a magnetic field in relationship to a moving point charge.

00:09 So first, let's discuss how to determine the direction at which the magnetic field is going.

00:15 So it says that we have a point charge, it has a magnitude of q, and it's moving in the positive y direction.

00:23 Okay.

00:25 And then we want to find the magnetic field at different points.

00:28 So in a, it asks us to find the magnetic field at 0 .5 in the x direction.

00:33 So let's call that like right here.

00:35 Because imagine that this is the y direction, this is the x direction, and we are looking for the magnetic field here.

00:43 So the direction, if you take your thumb and you point your thumb in the direction in which the charge is moving, so that would be here.

00:51 So you point your thumb in the plus y direction or straight vertically.

00:59 And then curl your fingers, the way you're curling your fingers is the way that the magnetic field is going.

01:07 So in this case, so you point your thumb and up, and then you curl your fingers, and if you're curling your fingers and looking for the magnetic field here, you'll notice that it's going into the screen.

01:18 Right.

01:19 So we would call that the negative z direction.

01:21 If you were looking for it here, you point your thumb in the positive y direction, and then you curl your fingers and you'll notice that it's coming out of the screen.

01:31 So we represent coming out of the screen or the positive z direction as a circle, and then we denote going into the screen or the negative z direction as an x.

01:44 Okay, so you use the right -hand rule, you point your thumb in the direction in which the charge is moving, and then you curl your fingers in the way that you curl your fingers at different points is the direction in which the magnetic field is going.

01:58 So that's how you find the direction.

02:00 To find the magnitude, we're going to use this formula, which is b equals mu not.

02:07 Muh is a constant, times q, which is the magnitude of our point charge, times v, which is the velocity.

02:16 So b and v both have a vector symbol above them because they have a direction.

02:21 They're not just a magnitude, but they have a direction, times sine of theta.

02:26 And theta is going to be the angle between the direction in which your point charge is moving and where your magnetic field is or where you're looking for it in relationship to that point charge.

02:40 So your theta in this example here is going to be 90 degrees, right? because it's moving in the positive y direction.

02:50 We're looking for the magnetic field in the positive x direction, so the angle between the two is 90.

02:57 And we're going to divide that by r squared and r is, sorry, by 4 pi r squared and r is going to be the distance between the two.

03:10 So if here's your point charge at the origin, it's going this way.

03:14 We're looking for the magnetic field here and it tells us we're looking for it at 0 .5, and 0 .5 is the distance between the two.

03:23 It's a straight line between where your point charge is located and where you're point.

03:27 You're looking for the magnetic field.

03:29 Okay, and we're just going to simplify this a little bit.

03:32 So, mu not is equal to 4 pi times 10 to the negative 7th times q times v times sine of theta divided by 4 pi r squared.

03:46 And you'll notice these two 4 pies cancel.

03:49 So we're left with 1 times 10 to the negative 7 times q times v times sine of theta over r squared.

03:58 So we're just simplified that a little bit.

04:01 Okay, so for the first part, here's our point charge, moving in the positive y direction.

04:05 We're looking for the magnetic field right here.

04:11 So our r is going to be 0 .5, as we said, and it's going to be going into the screen.

04:17 We call that the negative z direction.

04:19 So it's coming out of the screen, like when you curl your fingers and you're looking for it here, point your thumb in the direction in which the point chart is moving, curl your fingers, and if you were looking for it here, you'll notice that you're curling your fingers out of the screen.

04:32 So if you're curling them out of the screen, we represent that as a zero, or sorry, a circle, which would be the positive z direction.

04:41 And then if they're going into the screen, that would be marked with an x, and that is the negative z direction.

04:48 Okay.

04:49 So, and you'll notice the theta between these two is 90 degrees.

04:57 And like i said, it's going into the screen here.

04:59 And this is where our magnetic fields can be located for part a.

05:05 So we have 1 times 10 to the negative 7th times our magnitude or our charge, which is six micro -coolums.

05:14 You have to convert that to coolums.

05:15 So that would be 6 times 10 to the negative 6th times our velocity, which is 8, 8 times 10 to the 6 meters per second, times sine of theta.

05:29 And theta is 90, as i just said, for part a, and then divide that by r squared, which i said was 0 .5, because this is the distance between the two, and this is the angle.

05:41 And b is going to be 1 .92 times 10 to the negative 5th...

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