Magnetic Field at the Center of a Wire Loop

In summary: If you integrate correctly, you should get an expression for the field at the center due to the entire loop. In summary, we are trying to find the z component of the magnetic field at the center of a circular loop with radius r, through which a steady current I is flowing in a counterclockwise direction. We use the Biot-Savart law, which states that the magnetic field at a point is equal to the integral of the magnetic field due to each small element of the current around the loop. After taking into account the vector product, we get the expression d\vec{B} = \frac{\mu_0 I}{4 \pi r^2}\;d\ell, which we then integrate around
  • #1
cse63146
452
0

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.

Homework Equations





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.
 
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  • #2
cse63146 said:

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.

Homework Equations





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]
 
  • #3
Biot-Savart law

cse63146 said:
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.)
 
  • #4
Isnt the equation I posted the Biot-Savart law?
 
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  • #5
cse63146 said:
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.
 
  • #6
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?
 
  • #7
cse63146 said:
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.
 
  • #8
Doc Al said:
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?
 
  • #9
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!)
 
  • #10
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?
 
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  • #11
cse63146 said:
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.
 

Related to Magnetic Field at the Center of a Wire Loop

1. What is the magnetic field at the center of a wire loop?

The magnetic field at the center of a wire loop is zero. This is because the magnetic field lines created by the current flowing through the loop are circular and cancel each other out at the center.

2. How does the shape of the wire loop affect the magnetic field at the center?

The shape of the wire loop does not affect the magnetic field at the center. As long as the wire loop is circular, the magnetic field will always be zero at the center.

3. What happens to the magnetic field at the center of a wire loop if the current is increased?

If the current through the wire loop is increased, the magnetic field at the center will also increase. This is because the strength of the magnetic field is directly proportional to the current.

4. Can the magnetic field at the center of a wire loop be changed?

No, the magnetic field at the center of a wire loop cannot be changed. It will always be zero as long as the loop is circular and the current remains constant.

5. How is the direction of the magnetic field at the center of a wire loop determined?

The direction of the magnetic field at the center of a wire loop is determined by the direction of the current flowing through the loop. The field lines will always be in a counterclockwise direction when viewed from above the loop.

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