Calculus relationship between current, resistance, and voltage

In summary, Ohm's law states that voltage is proportional to current and resistance. If you want to change the direction of the current, you need to change the direction of the voltage.
  • #1
partialfracti
22
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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?
 
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  • #2
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
 
  • #3
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.
 
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  • #4
partialfracti said:
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?

[tex] I= \int_S \vec J \cdot d\vec S ,\;\;\;\; V= -\int_C \vec E \cdot d\vec l [/tex]

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

[tex] R=\frac{-\int_C \vec E \cdot d\vec l }{\int_S \vec J \cdot d\vec S} [/tex]

Which is a fancy way of saying

[tex]R=\frac V I [/tex]

:rofl: :rofl:

Or if you still want more:

[tex] 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 [/tex]

Where [itex]\sigma [/itex] is conductance, [itex]\mu [/itex] is mobility and [itex]\rho_v [/itex] is volume charge density.
 
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  • #5
Thats so right, nice to see the information
 
  • #6
yungman said:
[tex] I= \int_S \vec J \cdot d\vec S ,\;\;\;\; V= -\int_C \vec E \cdot d\vec l [/tex]

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

[tex] R=\frac{-\int_C \vec E \cdot d\vec l }{\int_S \vec J \cdot d\vec S} [/tex]

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?
 
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  • #7
partialfracti said:
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.
 

1. What is the relationship between current, resistance, and voltage?

The relationship between current, resistance, and voltage is described by Ohm's Law, which states that the current flowing through a conductor is directly proportional to the voltage and inversely proportional to the resistance. This means that as resistance increases, current decreases, and as voltage increases, current increases.

2. How do I calculate current, resistance, and voltage using calculus?

To calculate current, resistance, and voltage using calculus, you can use the formulas I = dQ/dt, R = dV/dI, and V = IR, where I is current, Q is charge, t is time, V is voltage, and R is resistance. These formulas can be derived using the principles of calculus, such as differentiation and integration.

3. What is the significance of the slope of a current-voltage graph?

The slope of a current-voltage graph represents the resistance of a conductor. The steeper the slope, the higher the resistance. This is because resistance is equal to the ratio of voltage to current, and the slope of a graph is the change in voltage divided by the change in current, which is equal to resistance.

4. How does calculus help us understand the behavior of electrical circuits?

Calculus helps us understand the behavior of electrical circuits by allowing us to model and analyze the relationships between current, resistance, and voltage. By using calculus, we can calculate the current, resistance, and voltage at any point in a circuit, as well as predict how these values will change over time. This allows us to design and optimize circuits for different purposes.

5. Can calculus be used to solve real-life problems related to current, resistance, and voltage?

Yes, calculus can be used to solve real-life problems related to current, resistance, and voltage. For example, it can be used to determine the optimal resistance for a circuit to achieve a desired current or voltage, or to analyze the behavior of complex circuits with multiple components. Calculus is a powerful tool in understanding and solving real-world problems in the field of electrical engineering.

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