Calculus relationship between current, resistance, and voltage

[partialfracti]

I remember that Position, velocity, and acceleration are all related in calculus somehow. Perhaps if one differentiates position, the result is the velocity, and if one differentiates velocity, the result is the acceleration. And the process can be reversed by integration. In this case, perhaps it would be that if one integrates the acceleration, one gets the velocity. And if one integrates velocity, one gets position.

I know about Ohm's Law that Current equals voltage divided by resistance.

In the field of electromagnetism in calculus, are current, resistance, and voltage related in a way analagous to the relationship between position, velocity, and acceleration in calculus? If so, what is the relationship of current, voltage, and resistance in terms of calculus?

 

[Redbelly98]

Welcome to Physics Forums.

They are not related in terms of calculus. Instead, it is a simple proportionality relation as given by Ohm' Law:

V = I R​

 

[Dr Lots-o'watts]

The V = RI relation is most useful in applied electrical engineering, such as when designing electrical circuits with ready-made components.

However, when studying the individual components of circuitry, i.e. when working the actual physics of the materials involved, Ohm's law is written in the alternative form E = rhoJ. In this form, you have the electric field E, and the current density J, which are vectors that can eventually be plugged into Maxwell's equations, and models of condensed matter, depending on the particular system studied.

And one can usually get as much calculus as their appetite can handle when they start using EM and condensed matter theory.

 

[yungman]

I remember that Position, velocity, and acceleration are all related in calculus somehow. Perhaps if one differentiates position, the result is the velocity, and if one differentiates velocity, the result is the acceleration. And the process can be reversed by integration. In this case, perhaps it would be that if one integrates the acceleration, one gets the velocity. And if one integrates velocity, one gets position.

I know about Ohm's Law that Current equals voltage divided by resistance.

In the field of electromagnetism in calculus, are current, resistance, and voltage related in a way analagous to the relationship between position, velocity, and acceleration in calculus? If so, what is the relationship of current, voltage, and resistance in terms of calculus?

$$I= \int_S \vec J \cdot d\vec S ,\;\;\;\; V= -\int_C \vec E \cdot d\vec l$$

Resistor...well is resistor! If you don't like V=IR then resistor is:

$$R=\frac{-\int_C \vec E \cdot d\vec l }{\int_S \vec J \cdot d\vec S}$$

Which is a fancy way of saying

$$R=\frac V I$$



Or if you still want more:

$$I= \int_S \vec J \cdot d\vec S \;=\; \int_S \sigma \vec E \cdot d\vec S \;=\; \int_S \mu\rho_v \vec E \cdot d\vec S$$

Where $\sigma$ is conductance, $\mu$ is mobility and $\rho_v$ is volume charge density.

 

[ukbiker]

Thats so right, nice to see the information

 

[partialfracti]

$$I= \int_S \vec J \cdot d\vec S ,\;\;\;\; V= -\int_C \vec E \cdot d\vec l$$

Resistor...well is resistor! If you don't like V=IR then resistor is:

$$R=\frac{-\int_C \vec E \cdot d\vec l }{\int_S \vec J \cdot d\vec S}$$

What is dl with an arrow over the l? What is dS with an arrow over the S?

I don't think that the C next to the integration sign means current since I usually means current. What does the C mean next to the integration sign?

 

[yungman]

What is dl with an arrow over the l? What is dS with an arrow over the S?

I don't think that the C next to the integration sign means current since I usually means current. What does the C mean next to the integration sign?

C is for line integral, S is for surface integral.

http://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtI.aspx

I like Paul Dawnkins book/notes. Serve on that site and find surface integral. You can even download the whole book.