Physical significance of dot and cross products in electrodynamics

In summary: The divergence of a vector field is a scalar field defined by\mathrm{div} \vec{V}=\vec{\nabla} \cdot \vec{V}=\partial_i V^i.
  • #1
nouveau_riche
253
0
i know that del.v=divergence of vector v
and del x v=curl of vector v

can anyone justify for the same? how dot product is physically connected to divergence property?
 
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  • #2
It is just notational convenience. The formula for divergence looks like a dot product.
 
  • #3
If you have an Euclidean space (e.g., [itex]\mathbb{R}^3[/itex] that describes the spatial hypersurface of an observer in an inertial frame in both Newtonian and special-relativistic physics), the derivatives

[tex]\partial_i=\frac{\partial}{\partial x^i}[/tex]

act formaly like components of a co-vector (covariant components).

It's convenient to write this in vector notation as the nabla symbol [itex]\vec{\nabla}[/itex]. In Cartesian coordinates you can now easily play with that symbol to create new tensor fields out of given tensors by operating on their corresponding components wrt. to the Cartesian basis (and co-basis). The most important examples, all appearing in electromagnetism as well as fluid dynamics are

The Gradient

The gradient of a scalar field is a vector field (more precisely a one-form) and defined by

[tex]\mathrm{grad} f=\vec{\nabla} f=\vec{e}^j \partial_j f.[/tex]

The Divergence

The divergence of a vector field is a scalar field defined by

[tex]\mathrm{div} \vec{V}=\vec{\nabla} \cdot \vec{V}=\partial_i V^i.[/tex]


The Curl

The curl of a vector field is given by

[tex]\mathrm{curl} \vec{V}=\vec{\nabla} \times \vec{V}.[/tex]

The structure of this becomes more clear by looking at it in terms of alternating differential forms (Cartan calculus). Given a 1-form (co-vector) [itex]\omega = \mathrm{d}x^j \omega_j[/itex], you get an alternating 2-form by

[tex]\mathrm{d} \omega = \mathrm{d} x^j \wedge \mathrm{d} x^k \partial_j \omega_k = \frac{1}{2} \mathrm{d} x^j \wedge \mathrm{d} x^k (\partial_j \omega_k-\partial_k \omega_j).[/tex]

In more conventional terms, you have a antisymmetric rank-2 tensor, given by its covariant components [itex]\partial_j \omega_k-\partial_k \omega_j[/itex]. In 3 dimensions, you can map this uniquely to a vector field by hodge dualization:

[tex][\mathrm{curl} \vec{V}]^j=\epsilon^{jkl} (\partial_k V_l-\partial_j V_k).[/tex]

Further, since in Euclidean spaced the metric has components [tex]\delta_{jk}[/tex] wrt. Cartesian Coordinates, you have [tex]V^j=V_j[/tex].
 
  • #4
DaleSpam said:
It is just notational convenience. The formula for divergence looks like a dot product.

can u justify with an example?
 
  • #5
Recall
[tex](a,b,c)\cdot(d,e,f) = ad+be+cf[/tex]
[tex]\text{Divergence}(f)=\frac{\partial}{\partial x}f_x+\frac{\partial}{\partial y}f_y+\frac{\partial}{\partial z}f_z=\left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \cdot \left( f_x, f_y, f_z \right) = \nabla \cdot f[/tex]
 
  • #6
DaleSpam said:
Recall
[tex](a,b,c)\cdot(d,e,f) = ad+be+cf[/tex]
[tex]\text{Divergence}(f)=\frac{\partial}{\partial x}f_x+\frac{\partial}{\partial y}f_y+\frac{\partial}{\partial z}f_z=\left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \cdot \left( f_x, f_y, f_z \right) = \nabla \cdot f[/tex]

how is the left hand side of your second equation, a divergence?
 
  • #7
[tex]\nabla.\phi = \frac{\partial \phi_x}{\partial x} + \frac{\partial \phi_y}{\partial y}+\frac{\partial \phi_z}{\partial z}[/tex]

It looks like the dot product of [tex]\frac{\partial}{\partial x} +\frac{\partial}{\partial y} + \frac{\partial}{\partial z}[/tex] and [tex](\phi_x,\phi_y,\phi_z)[/tex].

It's an abuse of notation.
 

1. What is the dot product in electrodynamics and what is its physical significance?

The dot product in electrodynamics is a mathematical operation that takes two vectors and produces a scalar value. In terms of physical significance, the dot product is used to calculate the amount of work done by a force in a given direction. In electrodynamics, it is used to determine the component of a force acting in a certain direction, which is important in understanding the motion of charged particles in electric and magnetic fields.

2. How does the dot product relate to electric fields in electrodynamics?

The dot product is used to calculate the electric potential energy in a given electric field. It is also used to calculate the electric flux, which is a measure of the amount of electric field passing through a surface. In both cases, the dot product is used to find the component of the electric field in a specific direction.

3. What is the physical significance of the cross product in electrodynamics?

The cross product in electrodynamics is a mathematical operation that takes two vectors and produces a third vector that is perpendicular to the original two. In terms of physical significance, the cross product is used to calculate the torque on a charged particle in a magnetic field. It is also used to determine the direction of the magnetic force acting on a moving charged particle.

4. How is the cross product used in calculating magnetic fields in electrodynamics?

The cross product is used to calculate the magnetic field created by a current-carrying wire or a moving charged particle. It is also used to determine the magnetic flux, which is a measure of the amount of magnetic field passing through a surface. In both cases, the cross product is used to find the direction of the magnetic field in a specific location.

5. What is the relationship between the dot and cross products in electrodynamics?

The dot and cross products are both mathematical operations used in electrodynamics, but they have different physical meanings. The dot product is used to find the component of a vector in a given direction, while the cross product is used to find a perpendicular vector. However, in some cases, the dot and cross products may be used together to find the angle between two vectors or to calculate the work done by a force in a magnetic field.

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