Integral of magnetic field over the sphere

In summary, there are two equations that relate magnetic field (B) and current (I) within a sphere with radius R. The first equation, derived in Jackson, is \int B \,dV= 2/3\mu_0 M, where M is the magnetic moment of all the currents. The second equation is \int B \,dV= 4/3 \pi R^3 B(0) when all the current is outside the sphere, with B(0) being the magnetic field at the center of the sphere. It is questioned whether these two equations hold simultaneously when all the currents are on the surface of the sphere. The solution is to work it out and see.
  • #1
hokhani
504
8
If all the currents were inside a sphere with the radius R, then we would have [itex]\int B \,dV= 2/3\mu_0 M[/itex] where [itex]M[/itex] is magnetic moment of all the currents and [itex]B[/itex] is magnetic field. If all the current were outside the sphere, then we would have[itex]\int B \,dV= 4/3 \pi R^3 B(0) [/itex] where [itex]B(0)[/itex]is magnetic field at center of the sphere (Both the relations above are derived in Jackson).
Now, how about the situation in which all the currents were on the surface of the sphere? Can one say that both the two relations above are hold simultaneously?
 
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  • #2
Why not work it out and see?
 

FAQ: Integral of magnetic field over the sphere

What is the integral of magnetic field over the sphere?

The integral of magnetic field over the sphere is a mathematical calculation that determines the total magnetic flux passing through a closed surface in three-dimensional space. It is an important concept in electromagnetism and is often used to calculate the strength and behavior of magnetic fields.

How do you calculate the integral of magnetic field over the sphere?

To calculate the integral of magnetic field over the sphere, you must first determine the magnetic field at each point on the surface of the sphere. Then, you can use the formula for the surface integral to sum up the magnetic field values over the entire surface. This results in the total magnetic flux passing through the sphere.

What is the significance of the integral of magnetic field over the sphere?

The integral of magnetic field over the sphere is significant because it allows us to understand and quantify the behavior of magnetic fields in three-dimensional space. This is important for many applications, such as designing electromagnets and understanding the behavior of Earth's magnetic field.

Does the integral of magnetic field over the sphere have any real-world applications?

Yes, the integral of magnetic field over the sphere has many real-world applications. For example, it is used in the design of MRI machines, which use magnetic fields to create images of the body's internal structures. It is also used in geophysics to map variations in Earth's magnetic field, which can provide valuable information about the planet's structure and composition.

What factors can affect the value of the integral of magnetic field over the sphere?

The value of the integral of magnetic field over the sphere can be affected by the strength and direction of the magnetic field, as well as the size and shape of the sphere. Other factors such as the electrical conductivity of the material inside the sphere and the presence of other magnetic fields can also influence the result.

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