Solving Constant Magnetic Field Vector Problem: Annoying Vectors

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In summary, for a constant magnetic field, the vector potential is equal to half the cross product of the magnetic field and position vector. By using the vector identity for the curl of a cross product, we can see that the second term in the equation for the curl of the vector potential vanishes because the gradient of a constant vector field is zero. This can also be shown using index notation.
  • #1
latentcorpse
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for a constant magnetic field [itex]\vec{B}[/itex] everywhere, [itex]\vec{A}=\frac{1}{2} \vec{B} \times \vec{r}[/itex]

because (im not going to use vector notation to save time)

[itex]\nabla \times (B \times r)= (\nabla \cdot r)B + (r \cdot \nabla)B - (\nabla \cdot B)r - (B \cdot \nabla)r[/itex]

the first term gives 3B
the third term vanishes
the fourht term gives -B
so to get the answer i need the second term to vanish but i can't get it to go away - how do i do this?
 
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  • #2
The second term vanishes because the gradient of a constant vector field is zero. None of the components are changing, so their space derivatives and hence the gradient is zero.
 
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  • #3
how can you take the gradient of a vector - you need index notation yes?

consider [itex](m \cdot \nabla)r[/itex]
is this just [itex]m_j \partial_j r_i=m_i[/itex]
 
  • #4
The gradient of a vector (a,b,c) just (grad a, grad b, grad c).
 
  • #5
but [itex]\nabla \phi=(\frac{\partial \phi}{\partial x},\frac{\partial \phi}{\partial y},\frac{\partial \phi}{\partial z})[/itex] so doesn't that give us a nine component vector - is that just a tensor?

e.g. [itex]\partial_i r_j =\delta_{ij}[/itex]?
 
  • #6
Sorry, I misread the equation. The second term is [tex] (r\cdot\nabla) B [/tex], which is

[tex] \sum_{i = 1}^3 (r_x \partial_x + r_y \partial_y + r_z \partial_z) B_i \mathbf{e}_i[/tex]
 
  • #7
how is that 0 though?

why can't this be done using eisntein summation convention i.e.
[itex]r_j \partial_j B_i=...[/itex] i can't get it to go any further?
 
  • #8
The derivatives of [tex] B_i [/tex] are zero, so what you just wrote must be zero.
 

1. What is a constant magnetic field vector?

A constant magnetic field vector is a vector that represents the strength, direction, and location of a magnetic field that remains the same over time and space.

2. Why are vectors important in solving constant magnetic field problems?

Vectors are important because they allow us to represent the magnitude and direction of the magnetic field in a visual and mathematical way, making it easier to solve complex problems and analyze the behavior of the magnetic field.

3. What are the steps involved in solving a constant magnetic field vector problem?

The steps involved in solving a constant magnetic field vector problem include identifying the given information, determining the direction and magnitude of the magnetic field vector, calculating the resultant vector, and using the right-hand rule to determine the direction of the magnetic field.

4. What is the right-hand rule and how is it used in solving constant magnetic field vector problems?

The right-hand rule is a method used to determine the direction of the magnetic field. It states that if you point your right thumb in the direction of the current, and your fingers in the direction of the magnetic field, your palm will point in the direction of the force experienced by a positive charge. In solving constant magnetic field vector problems, the right-hand rule is used to determine the direction of the magnetic field from the given information.

5. What are some common challenges in solving constant magnetic field vector problems?

Some common challenges in solving constant magnetic field vector problems include understanding the concept of vectors, knowing how to use the right-hand rule, and identifying the correct formula to use based on the given information. It is also important to pay attention to units and ensure that all calculations are done correctly.

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