Non-differential Form of Biot-Savart Law: Comparison and Confusion Clarified

[Chemist@]

What is the non-differential form of the Biot-Savart law? Is it:
B=mi*I/(4R*pi)*(cos(a)-cos(b)) or B=mi*I/(4R*pi)*(cos(a)+cos(b))?

For a infinitely long conductor, the law is:
B=mi*I/(2R*pi) because a=0 and b=pi. So I would say that the correct expression is the one where the cos are subtracted, but I was solving a problem where a point is in the center of a square-like conductor and they used the formula with the addition of cos to get the magnetic field at that point, as with the subtraction the answer would be zero. What equation is the correct one, I am really confused?

 

[mfb]

The Biot-Savart law is an integral along the current flow, therefore the integral will depend on the geometry of your current flow. Also, what are "a" and "b"?

Laplace?

 

[ScienceAlum]

As mfb mentioned, Biot-Savart Law is dependent on the geometry of the current carriers. The cosines might stem from the fact that there is a cross product in numerator of the Biot-Savart Law. The cross product, between the length of the conductor and the unit vector from the current to the field, or:

$$ \vec{ds} \times \hat{r} $$

Can also be expressed as:

$$ |\vec{ds}| |\hat{r}| cos \theta $$

This could be the reason you were dealing with cosines.

To get a better understanding of the Biot-Savart formula and any other formula in Electricity & Magnitude I'd suggest using:

http://theeqns.com/introduction-to-electricity--magnetism-with-calculus.html

 

[jtbell]

So I would say that the correct expression is the one where the cos are subtracted, but I was solving a problem where a point is in the center of a square-like conductor and they used the formula with the addition of cos to get the magnetic field at that point

Are the angles defined the same way in both cases?

 

[pmacias]

The non-differential form of the Biot-Savart law is a mathematical expression that describes the magnetic field produced by a steady current in a conductor. It is given by B=mi*I/(4R*pi)*(cos(a)-cos(b)), where mi is the permeability of the medium, I is the current, R is the distance from the current to the point of interest, a is the angle between the current and the distance vector, and b is the angle between the current and the normal vector to the plane containing the current and the point of interest.

In the case of an infinitely long conductor, the non-differential form simplifies to B=mi*I/(2R*pi), as a=0 and b=pi. This means that the magnetic field at any point along the conductor is the same, and is directly proportional to the current and inversely proportional to the distance from the conductor.

In your specific case, where the point of interest is in the center of a square-like conductor, the correct expression to use is the one with the addition of cos, B=mi*I/(4R*pi)*(cos(a)+cos(b)). This is because in this scenario, the angles a and b are not equal to 0 and pi, and thus both terms must be taken into account in order to accurately calculate the magnetic field at the point.

It is important to note that the Biot-Savart law is a vector equation, meaning that the direction of the magnetic field must also be taken into account. In the case of a square-like conductor, the magnetic field will have different components in the x and y directions, and the correct expression must be used to accurately calculate both components.

In summary, the correct non-differential form of the Biot-Savart law to use depends on the specific scenario and the values of the variables involved. It is important to carefully consider all relevant factors and choose the appropriate expression for accurate calculations.