Help finding the magnetic field of a current loop

In summary: Ah, okay now I understand. The cross product does not show that the dB is only on the z axis so I must make that change myself. Once I took this into effect then my equation equaled that equation which I was supposed to have, many thanks!
  • #1
Boltzman Oscillation
233
26
Homework Statement
Find the magnetic field at a point due to a loop of wire.
Relevant Equations
$$ db = \frac{\vec{I}\vec{dl} \times \vec{r}}{cr^3}$$
Hi all, my work is shown on the attached image. The boxed equation is what I get to but I do not understand how to go from there to what the book has. I am guessing that the problem arises when trying to solve the cross product. I understand that I will need to find the value of the sine of the angle between the arguments of the cross product but I do not see how I could do that, all I know is that the direction will be in the z axis. Any help is appreciated as always.
16369316997682310877457715037074.jpg
 
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  • #2
Theta is not the angle between the vectors ##\vec{d l}, \vec r##. What is the angle between them?
 
  • #3
haruspex said:
Theta is not the angle between the vectors ##\vec l, \vec r##. What is the angle between them?
Yes, theta is not the angle between the vectors dl and r, I forgot to label the angle on the drawing. I think I can assume that angle to be close to 90° if the point is very far away but that would still not get me the answer that was provided.
 
  • #4
Boltzman Oscillation said:
Yes, theta is not the angle between the vectors dl and r, I forgot to label the angle on the drawing. I think I can assume that angle to be close to 90° if the point is very far away but that would still not get me the answer that was provided.
It is not just approximately 90°, it is exactly 90°, regardless of the distance to the point. E.g. consider the point at the centre of the circle.

I'm not following how you took the z axis component of the field. Looks like you forgot to multiply by the sine of the apex half angle.
 
  • #5
haruspex said:
It is not just approximately 90°, it is exactly 90°, regardless of the distance to the point. E.g. consider the point at the centre of the circle.

I'm not following how you took the z axis component of the field. Looks like you forgot to multiply by the sine of the apex half angle.
Im sorry but what is the apex half angle? From what I've gathered so far my current equation is
$$\frac{I*a*d \theta *r}{r^3}\vec{d}$$

I would somehow have to show that r = a which it isn't so I know somewhere a long the way I did something wrong.

I thought that by solving the cross product then I would get the z axis component. Hence:

$$\vec{I}\vec{dl} \times \vec{r} = I*dl*r*sin(\theta)\vec{z} $$
 
  • #6
haruspex said:
It is not just approximately 90°, it is exactly 90°, regardless of the distance to the point. E.g. consider the point at the centre of the circle.

I'm not following how you took the z axis component of the field. Looks like you forgot to multiply by the sine of the apex half angle.
Ah, okay now I understand. The cross product does not show that the dB is only on the z axis so I must make that change myself. Once I took this into effect then my equation equaled that equation which I was supposed to have, many thanks!
 

1. What is a current loop?

A current loop is a closed circuit through which an electric current flows. It can be in the form of a wire bent into a loop or a coil of wire.

2. How is the magnetic field of a current loop calculated?

The magnetic field of a current loop can be calculated using the formula B = μ₀I/2r, where B is the magnetic field, μ₀ is the permeability of free space, I is the current in the loop, and r is the distance from the center of the loop to the point where the magnetic field is being measured.

3. What factors affect the magnetic field of a current loop?

The magnetic field of a current loop is affected by the strength of the current, the size and shape of the loop, and the distance from the loop to the point where the magnetic field is being measured.

4. How can the direction of the magnetic field of a current loop be determined?

The direction of the magnetic field of a current loop can be determined using the right-hand rule. If you point your thumb in the direction of the current, your fingers will curl in the direction of the magnetic field.

5. Can the magnetic field of a current loop be changed?

Yes, the magnetic field of a current loop can be changed by changing the current in the loop, altering the shape or size of the loop, or changing the distance from the loop to the point where the magnetic field is being measured.

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