Positive point charges q=+8.00 μC and q^'=+3.00 μC are moving relative to an observer at point P

Positive point charges q=+8.00 μC and q^'=+3.00 μC are...

Positive point charges $q=+8.00 \mu \mathrm{C}$ and $q^{\prime}=+3.00 \mu \mathrm{C}$ are moving relative to an observer at point $P$, as shown in Fig. $\mathbf{E} 28.6$. The distance $d$ is $0.120 \mathrm{~m}, v=4.50 \times 10^{6} \mathrm{~m} / \mathrm{s},$ and $v^{\prime}=9.00 \times 10^{6} \mathrm{~m} / \mathrm{s} .$ (a) When the two charges are at the locations shown in the figure, what are the magnitude and direction of the net magnetic field they produce at point $P ?$ (b) What are the magnitude and direction of the electric and magnetic forces that each charge exerts on the other, and what is the ratio of the magnitude of the electric force to the magnitude of the magnetic force? (c) If the direction of $\overrightarrow{\boldsymbol{v}}^{\prime}$ is reversed, so both charges are moving in the same direction, what are the magnitude and direction of the magnetic forces that the two charges exert on each other?

Positive point charges $q=+8.00 \mu \mathrm{C}$ and $q^{\prime}=+3.00 \mu \mathrm{C}$ are
moving relative to an observer at point $P$, as shown in Fig. $\mathbf{E} 28.6$. The distance $d$ is $0.120 \mathrm{~m}, v=4.50 \times 10^{6} \mathrm{~m} / \mathrm{s},$ and
$v^{\prime}=9.00 \times 10^{6} \mathrm{~m} / \mathrm{s} .$ (a) When the two
charges are at the locations shown in the figure, what are the magnitude and direction of the net magnetic field they produce at point $P ?$
(b) What are the magnitude and direction of the electric and magnetic forces that each charge exerts on the other, and what is the ratio of the magnitude of the electric force to the magnitude of the magnetic force? (c) If the direction of $\overrightarrow{\boldsymbol{v}}^{\prime}$ is reversed, so both charges are moving in the same direction, what are the magnitude and direction of the magnetic forces that the two charges exert on each other?

Key Concepts

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field produced by a moving charge or the direction of the magnetic force acting on a charge in a magnetic field. When dealing with cross products in electromagnetism, it is essential for correctly establishing the orientation of the resulting vectors.

Lorentz Force Law

The Lorentz force law combines the effects of electric and magnetic fields on a charged particle, stating that the net force acting on a moving charge is the sum of the electric force (qE) and the magnetic force (q(v x B)). This law is critical for analyzing the forces acting on charges when they are in motion in the presence of both electric and magnetic fields.

Superposition Principle

The superposition principle in electromagnetism states that the total field at any point is the vector sum of the individual fields produced by each charge or current element. This concept allows for the independent calculation of electric and magnetic fields from multiple sources, which can then be combined to determine net forces or fields.

Magnetic Field of a Moving Charge

A moving electric charge creates a magnetic field in the surrounding space, described by the Biot–Savart law. The magnetic field at any point is proportional to the charge’s magnitude and its velocity, and its direction is given by the cross product of the velocity vector and the unit vector pointing to the field point. This concept underlies the analysis of how charges in motion produce magnetic fields that interact with other moving charges.

Coulomb’s Law

Coulomb’s law governs the electrical interaction between stationary charges, stating that the electric force between two point charges is proportional to the product of their charges and inversely proportional to the square of the distance between them. It provides the foundation for understanding the electric forces in systems of point charges, regardless of their motion.

Transcript

00:01 Okay, problem 28 .6, we have two point charges of positive charge, of certain values, and they're moving in different directions, and there's certain distance apart from this point p.

00:14 So first question is when two charges are locations shown a figure, where the magnitude direction of the magnetic field they produce at point p.

00:22 Okay, so we're going to start off with writing the magnetic field of a moving point charge.

00:27 So we have b and then at.

00:32 Point p we're going to write is equal to mu not over four pi and then i'm going to write down one over distance squared and that's going to be similar between the two different terms so we have a term that's for the positive charge q and one for positive charge q prime and then in brackets i'm going to put down what the difference between those terms are because it's the same we're going to write on what's different and then they're just going to be added together so one is is going to be the charge times the vector v cross r hat.

01:10 So for the first one, we have qv, and then it's going to be the unit vector i hat cross with negative to j hat.

01:22 So i hat, cross with negative j hat.

01:26 That's the unit vector for the velocity crossed with the r hat vector extending from point to point q to be.

01:36 And then that's going to be added to q prime, v prime, charge, velocity, and it's going to be negative i -hat, it's going left, cross with positive j hat.

01:49 So negative i hat, cross with j hat.

01:57 Okay, so then that's going to then be equal to the same constants, v0 over for pi over d squared, and it's going to be the numbers we have here so it's going to be 8.

02:14 Now i'm just going to read this.

02:17 So q, v, i hat cross negative j hat is going to be negative k hat.

02:27 And then plus q prime, v prime.

02:32 And then we have negative i hat cross j hat.

02:35 That's going to be again negative k hat.

02:41 Okay, so then all this is numbers.

02:44 And then we have the negative k hat for each of them.

02:46 So if you solve that, we get four, three, three, 7 .5 micro tesla and a negative k -hat direction.

03:00 Okay, so that's part a.

03:04 Part b was magnitude direction of electric and negative fields.

03:10 Excerts on each other.

03:13 Okay, so first off, i'm going to start with the next page.

03:21 So we have the force, the electric field of key.

03:27 Acting on q prime that's going to be one over four pi epsilon not and then q prime q over 2d square two d is a distance that separates them q prime to q this is just coombs law okay so then that's just all calculate out that's all numbers we have so 3 .74 new and then we have to figure out what the direction is and that's going to be negative j hat.

04:09 So if we go back to the first page, q acting on q prime, those are both positive charges.

04:15 It's going to repel q prime is going to be pushed down, which is negative j hat.

04:20 Okay, and then due to symmetry in the calculation, the force of the electric field, q prime, acting on q, it's just going to switch those to q and q prime, but that's going to be the same value.

04:43 So then we just get 3 .74 newtons and a positive jad direction.

04:52 So the, oops, so the top charge is going to be repelled, push up.

04:59 Okay, so how about the next one? so force of due to the b field of q acting on q prime...

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