Gradient, Electric Potential, and Electric Field

In summary, gradient, electric potential, and electric field are all related concepts in the field of electromagnetism. The gradient is a mathematical tool used to describe the change in a physical quantity over a given distance. Electric potential is a scalar quantity that describes the potential energy of a charged particle in an electric field. Electric field, on the other hand, is a vector quantity that describes the force experienced by a charged particle in an electric field. Together, these concepts help us understand and predict the behavior of electrically charged particles in various scenarios, such as in circuits and electromagnetic interactions.
  • #1
KvGroOve
Hi guys!

I was wondering about the relation between the Gradient, Electric Potential, and Electric Field. I know that if you take the Gradient of a scalar field, you get a resultant vector field in which the vector points in the direction of greatest increase when you take a infinitesimally small step.

1) What if there are multiple directions which all offer the greatest yet same amount of increase in the value of the scalar function? I believe that if it's a local minimum, the gradient of that specific spot is 0. But what if it's not a local minimum? How does the gradient operator handle that sort of situation? (I'm guessing that would mean a cross-section of the scalar field would trace out an inflection point).

2) The Electric Field can be given by
E=-∇φ​
So from my understanding, the negative sign just makes the vector point in the opposite direction, which in this case, is the direction opposite to the greatest increase of Electric Potential. Is this direction guaranteed to be the direction of the greatest possible decrease of the Electric Potential? If so, it's not intuitive to me that the direction of greatest possible decrease is opposite of the direction of greatest possible increase. And again, what if there are multiple directions that offer the greatest possible decrease in Electric Potential?

Thanks!
 
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  • #2
KvGroOve said:
1) What if there are multiple directions which all offer the greatest yet same amount of increase in the value of the scalar function? I believe that if it's a local minimum, the gradient of that specific spot is 0. But what if it's not a local minimum? How does the gradient operator handle that sort of situation? (I'm guessing that would mean a cross-section of the scalar field would trace out an inflection point).
This does not happen. As long as the function can be well approximated by a linear function, there will always be a unique direction of fastest increase.

If the gradient is zero, then the linear part of the change in the function value vanishes and the change in the function is of second order or higher in the displacement.

KvGroOve said:
2) The Electric Field can be given by
E=-∇φSo from my understanding, the negative sign just makes the vector point in the opposite direction, which in this case, is the direction opposite to the greatest increase of Electric Potential. Is this direction guaranteed to be the direction of the greatest possible decrease of the Electric Potential? If so, it's not intuitive to me that the direction of greatest possible decrease is opposite of the direction of greatest possible increase. And again, what if there are multiple directions that offer the greatest possible decrease in Electric Potential?

It is a linear approximation of the function near the point so yes, it is the direction of fastest decrease in potential. Again, this presupposes that the function can be approximated by a linear function near the point (think a plane in 3D). If this is not the case the function is not differentiable.
 
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1. What is the difference between gradient, electric potential, and electric field?

The gradient is a mathematical concept that represents the rate of change of a function in a particular direction. Electric potential is a measure of the electrical potential energy per unit charge at a given point in space. Electric field is a vector quantity that represents the strength and direction of the force exerted on a charged particle by other charged particles in its surroundings.

2. How is electric potential related to electric field?

Electric potential and electric field are closely related. The electric field is the negative gradient of the electric potential. In other words, the electric field at a point is equal to the negative of the rate of change of electric potential with respect to distance at that point.

3. How are gradient, electric potential, and electric field used in practical applications?

Gradient, electric potential, and electric field are used in many practical applications, such as in the design of electrical circuits, the analysis of electrically charged systems, and in engineering fields such as telecommunications and power systems. They are also important in understanding the behavior of particles in fields, such as in medical imaging technologies like MRI and PET scans.

4. How can I calculate the electric potential and electric field at a given point?

To calculate the electric potential at a point, you can use the equation V = kQ/r, where V is the electric potential, k is the Coulomb's constant, Q is the charge, and r is the distance from the charge. To calculate the electric field at a point, you can use the equation E = kQ/r^2, where E is the electric field and the other variables are the same as before.

5. Can you explain the concept of equipotential surfaces?

Equipotential surfaces are imaginary surfaces in space where all points have the same electric potential. This means that no work is required to move a charged particle along an equipotential surface. The electric field lines are always perpendicular to equipotential surfaces, and the closer the lines are together, the stronger the electric field is. Equipotential surfaces are useful in visualizing and understanding the behavior of electric fields.

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