# Bio-Savart Law, current through wire in semi circle

• arrowface
In summary, the conversation discusses the method for finding the magnetic field at point P, which is the center of a semicircular loop of radius b, due to a current flowing in a wire with straight sections on either side of the loop. The Biot-Savart law is used to calculate the magnetic field, with dl and r being the key variables. The conversation also addresses how to apply this method to a square that is cut in half, by breaking the wire into separate sections and considering the unit vectors i and j.
arrowface

## Homework Statement

I solved this problem. I just have a general question at the very end.

A current flows in a wire that has straight sections on either side of a semicircular loop of radius b. Find the mag and direction of the magnetic field B at point P(center of loop).

......... periods = mag field pointing out
....._____...... x's = mag field pointing in
.../xxxxxxx\.... radius = "b" from point P to any part in the semi circle.
...../xxxxxxxxx\........y
-->--->--| xxxxPxxxx|--->---->--.....|
xxxxxxxxxxxxxxxxxxxxxxxxxxxxx...z(out)|___x

Bio-Savart

## The Attempt at a Solution

B = μ/4pi ∫(I(dl x r ))/r^2

I started out by pulling out the constants and everything I knew. Since dl is in the same direction as the current, the magnetic field from the straight pieces of the wire does not contribute to the magnetic field at point P due to the angle between dl and r:

B = μI/(4pi*b^2) ∫(dl x r )

As the current goes around the semi circle, a perfect 90 degrees is maintained between dl and r so r cancels out( sin(90)= 1 )

B = μI/(4pi*b^2) ∫ dl

The next step is finding dl which is the sum of which dl travels around the semi circle. Since it is half of a circle it would be pi multiplied by the radius which is b:

B = μI*pi*b/(4pi*b^2) >>>>>> B = -μI/4b in the negative z direction due to the magnetic field.--------------
My Question
--------------

What if the semicircle was not a semicircle, but a square that was cut in half? How would I deal with finding dl then?

It sounds like your confusion has to do with computing the cross product $d\mathbf{\vec{l}}\times \mathbf{\vec{r}}$. I would suggest breaking the wire up into the three separate sections, for example, the integral for the longest part of the half-square would have $d\mathbf{\vec{l}}= dx\mathbf{\hat{x}}$, since the current of that segment lies parallel to the $\hat{\mathbf{x}}$ axis. Then you just need to compute the cross product. Think about what $\mathbf{\vec{r}}$ could be; you are integrating along the current in the wire that lies at $y=b/2$ and $z=0$, from $x=\left[-b/2,b/2\right]$. You just need to compute the cross-product of the two quantities $d\mathbf{\vec{l}}\times \mathbf{\vec{r}}$ once you have this.

arrowface said:
1. --------------
My Question
--------------

What if the semicircle was not a semicircle, but a square that was cut in half? How would I deal with finding dl then?

You would have to use the Biot-Savart law using vectors.

Set up an x-y coord. system with P at the origin. The square is bounded by (-a,0), (-a,2a), (a,2a) and (a,0). I use i and j for unit vectors.
Biot-Savart: dB = kI(dl x r)/r^3

For the left vertical part of your square (x = -a): dl = dy j and r = a i - y j. r = |r|.
Integrate from y = 0 to y = 2a.

For the horizontal stretch y = 2a, dl = dx i and r = -x i - 2a j. Integrate from x = -a to +a.
Etc. Get the idea?

## 1. What is the Bio-Savart Law?

The Bio-Savart Law is a mathematical equation that describes the magnetic field produced by a steady electric current through a wire. It was discovered by French physicist Félix Savart and later refined by Jean-Baptiste Biot.

## 2. How does the Bio-Savart Law relate to current through a wire?

The Bio-Savart Law states that the magnetic field produced by a steady current through a wire is directly proportional to the current, the length of the wire, and the sine of the angle between the wire and the point where the magnetic field is being measured.

## 3. What is a semi-circle in relation to the Bio-Savart Law?

In regards to the Bio-Savart Law, a semi-circle refers to the shape of the path taken by the current-carrying wire. This shape is often used in calculations to simplify the equation and make it easier to solve.

## 4. How is the Bio-Savart Law used in scientific research?

The Bio-Savart Law is often used in scientific research to calculate the magnetic field produced by electric currents in various systems. It has applications in a wide range of fields, including electromagnetism, astrophysics, and medical imaging.

## 5. What are some limitations of the Bio-Savart Law?

Some limitations of the Bio-Savart Law include its inability to accurately predict the magnetic field produced by changing currents or non-uniform current distributions. It also does not take into account factors such as the magnetic permeability of materials and the effects of nearby magnetic fields.

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