Why is B(x,t)•l - B(x+dx,t)•l in Maxwell's Eqn 34.16?

[sparkle123]

Why is eqn 34.16
B(x,t)•l - B(x+dx,t)•l instead of
B(x+dx,t)•l - B(x,t)•l ?
Thanks!

 

[dynamicsolo]

The line integration is being followed around the rectangle counter-clockwise, so the segment along which the magnetic field is B is parallel to the direction of integration, while the segment with ( B + dB ) is anti-parallel to the integration direction.

 

[I like Serena]

Hi again sparkle123!

It's a matter of convention: one of the many applications of the right hand rule.
Point your thumb in the direction of the positive y-axis, and your fingers give the direction that is by convention positive.
This is shown by the arrowheads in the loop.

So B(x,t) is taken as positive, while B(x+dx,t) is taken as negative.

Edit: overtaken by dynamicsolo. ;)

 

[sparkle123]

Thanks dynamicsolo and I like Serena! :)

 

[rwisz]

Maxwell's Eqn 34.16 is a vector equation that describes the behavior of electromagnetic fields. The equation is used to understand the relationship between the magnetic field (B) and the electric field (E) at a specific point in space and time. The equation is written as:

∇ x B = μ0ε0(∂E/∂t)

where ∇ x B represents the curl of the magnetic field, μ0 is the permeability of free space, and ε0 is the permittivity of free space.

In this equation, the term B(x,t)•l represents the magnetic field at a specific point (x,t) along a specific direction (l). Similarly, B(x+dx,t)•l represents the magnetic field at a point that is a small distance (dx) away from the original point (x,t) along the same direction (l).

The reason why the equation is written as B(x,t)•l - B(x+dx,t)•l instead of B(x+dx,t)•l - B(x,t)•l is because the curl of a vector field is defined as the limit of the difference quotient as the distance between two points approaches zero. In other words, we want to understand the behavior of the magnetic field at a specific point as we get closer and closer to that point.

By subtracting B(x+dx,t)•l from B(x,t)•l, we are essentially looking at the change in the magnetic field as we move from one point (x+dx,t) to another (x,t). This allows us to understand how the magnetic field is changing at that specific point and in that specific direction.

Therefore, the reason why B(x,t)•l - B(x+dx,t)•l is used in Maxwell's Eqn 34.16 is because it helps us to understand the behavior of the magnetic field at a specific point and in a specific direction as we approach that point. It is a fundamental aspect of vector calculus and is crucial in understanding the behavior of electromagnetic fields.