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Magnetic Field at the Center of a Wire Loop

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1. Homework Statement

A piece of wire is bent to form a circle with radius r. It has a steady current I flowing through it in a counterclockwise direction as seen from the top (looking in the negative z direction).

What is B_z(0), the z component of B at the center (i.e., x = y = z = 0) of the loop?

Express your answer in terms of I, r, and constants like mu_0 and pi.

2. Homework Equations



3. The Attempt at a Solution

I know this equation:

[tex]\frac{(\mu_0)I}{2(\pi)r}[/tex]

but there is a hint that says I need to find the Integrand.

Thank You.
 
743
4
1. Homework Statement

A piece of wire is bent to form a circle with radius r. It has a steady current I flowing through it in a counterclockwise direction as seen from the top (looking in the negative z direction).

What is B_z(0), the z component of B at the center (i.e., x = y = z = 0) of the loop?

Express your answer in terms of I, r, and constants like mu_0 and pi.

2. Homework Equations



3. The Attempt at a Solution

I know this equation:

[tex]\frac{(\mu_0)I}{2(\pi)r}[/tex]

but there is a hint that says I need to find the Integrand.

Thank You.
Integrate the magnetic field around the circular path of radius r.
[tex]\oint \vec B \cdot d\vec r = ?[/tex]
 

Doc Al

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Biot-Savart law

I know this equation:

[tex]\frac{(\mu_0)I}{2(\pi)r}[/tex]
That's the magnetic field from an infinite straight current-carrying wire.

Look up the Biot-Savart law. That will give you the field from a current element.

but there is a hint that says I need to find the Integrand.
Right. Once you have the field from a current element, you'll need to integrate around the entire loop. (Since you are only asked to find the field at the center of the loop--as opposed to some arbitrary location--the integral will turn out to be quite doable.)
 
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Isnt the equation I posted the Biot-Savart law?
 
Last edited:

Doc Al

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Isnt the equation I posted the Biot-Savart law?
No. As I said, the equation you posted is the field from a long current-carrying wire. Look up the Biot-Savart law.
 
452
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Sorry, about that, I was looking at the wrong equation in my book.

B = [tex]\frac{\mu_0}{4\pi}[/tex] [tex]\frac{q(v X r}{r^2}[/tex]

since its circular motion B = [tex]\frac{qmv}{r}[/tex] <=Would I need to ingetrate this equation?
 

Doc Al

Mentor
44,803
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Sorry, about that, I was looking at the wrong equation in my book.

B = [tex]\frac{\mu_0}{4\pi}[/tex] [tex]\frac{q(v X r}{r^2}[/tex]
The one you want is in terms of current:
[tex]d\vec{B} = \frac{\mu_0 I d\vec{\ell}\times \hat{r}}{4 \pi r^2}[/tex]

Figure out what that is for a point in the center of the loop, then integrate around the loop.

since its circular motion B = [tex]\frac{qmv}{r}[/tex] <=Would I need to ingetrate this equation?
Not relevant; No circular motion here.
 
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Figure out what that is for a point in the center of the loop, then integrate around the loop.
Would it be

[tex]\vec{B} = \frac{\mu_0 I d}{4 \pi r^2}[/tex]

and then integrate that?
 

Doc Al

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Almost. After taking care of the vector product, it would be:

[tex]d\vec{B} = \frac{\mu_0 I}{4 \pi r^2}\;d\ell[/tex]

Integrate that around the loop. (It's easy!)
 
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is the [tex]d \ell[/tex] distance*length or the derivative of length.

Then I would [tex]\oint \vec{B} dr[/tex] like Reshma said?
 
Last edited:

Doc Al

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is the [tex]d \ell[/tex] distance*length or the derivative of length.
Neither. [tex]d \ell[/tex] is an element of length around the circumference of the circle. (That should tip you off as to what the integral is. :wink:)

Then I would [tex]\oint \vec{B} dr[/tex] like Reshma said?
No. Integrate the expression I gave in the last post, which is the field at the center due to a small element of the current, over the complete loop.
 

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