Solving N-dipole Problem in the Heart: How to Calculate Potential Fields

[Carl140]

I have been trying to solve this problem without any success. Not homework, just curious how to solve this, I'm trying to deduce physical models. Honestly I have no idea
how to attack it, so I'd really appreciate if you could please provide some help.

This is electrodynamics applied to medical physics.

Suppose n-dipoles are colocated in the ventricular tissue of the heart with a certain
fixed horizontal depth, which we denote by x.
Assume all these n-dipoles have constant dipolar moment, i.e they are continuously distributed. Now imagine a system of coordinates and suppose the n-dipoles are located at the points (x,y) where x is the fixed horizontal depth and y is the height (vertical distance between the dipoles), y is varying.

How would you write the expression that describes the potential field generated by these n- dipoles? under the assumptions previously mentioned.

I know it's an integral, but don't know how to attack this problem.

Thanks in advance.

 

[quantoshake11]

at large distances you can just sum up all the dipole moments and treat it as one big dipole, i think.. Maybe you could just calculate the potential of each dipole and then sum up all the individual solutions, choosing an appropiate coordinate system (you could let a computer do this for you).

 

[charng]

The potential field generated by n-dipoles in the heart can be described by the superposition of the individual potential fields generated by each dipole. This can be written as an integral over the entire distribution of dipoles, with each dipole contributing to the overall potential field based on its position and dipole moment.

To calculate this potential field, you can use the principle of superposition, which states that the total potential at any point is equal to the sum of the potentials generated by each individual dipole. This can be expressed mathematically as:

Φ(x,y) = Σ Φi(x,y)

Where Φ(x,y) is the total potential at a point (x,y) and Φi(x,y) is the potential generated by the individual dipole at that point.

To calculate the potential generated by a single dipole, you can use the formula:

Φi(x,y) = k * (p * cosθ)/r^2

Where k is a constant, p is the dipole moment, θ is the angle between the dipole and the point (x,y), and r is the distance between the dipole and the point (x,y).

To calculate the total potential field at a point, you can integrate this formula over the entire distribution of dipoles, with each dipole contributing to the overall potential field based on its position and dipole moment. The integral can be written as:

Φ(x,y) = ∫∫ k * (p * cosθ)/r^2 dx dy

Where the limits of integration are determined by the distribution of dipoles.

I hope this helps you in solving the n-dipole problem in the heart. Remember to always consider the assumptions and limitations of your model when interpreting the results.