By Dr. Shahid Ali Yousafzai

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If I write $$y = x-1$$ Then the line will actually start from $$-1$$ but the slope will remain the same. All these lines will be parallel to each other.

But now if I write an equation that is $$y = 2x+1$$ It means that the slope is now 2 while the intercept is 1, for each value of x, y will have 2 more values.

So, when the slope changes, the straight line inclination changes but its shape remains the same.

The line at which x is having values but the y value is zero is called slope zero line. The line at which the x and y values are the same called slope 1 line and when y value changes but the x value remains, then the slope is undefined.

So now we will come to our laws. Let me first discuss the equation and then we will move how we should write this equation. The very first thing $$V = IR$$ which we call the Ohm’ s law.

Actually this is not a law, because law is having some other requirements. This is actually an empirical relation.

The Ohm’ s law is actually $$j =\sigma E$$ where j is the current density, $$\sigma$$ is the conductivity and E is the applied electrical field.

So this is actually $$j \propto E$$ and then $$j = constant \times E$$ and this constant is actually dependent upon the material which we call conductivity of a material. and $$\sigma =\frac 1 \rho$$ where $$\rho$$ is called resistivity of a material. Now when I write that $$j = \frac 1 \rho E$$ Here $$j = \frac 1 A$$ which is perpendicular to it, So, I can write $$\frac 1 A = \frac 1 \rho \frac V L$$ and then I can write that $$\frac { \rho L }{ A } I=V$$ I define this quantity which is $$\frac { \rho L }{ A }$$ as the resistance of a material. This implies $$V = IR$$ Now we have to distinguish between the $$\rho$$ and the R.

$$\rho$$ is called the resistivity of a material and this is the intrinsic property. You can recognize a material from this property. While R is not an intrinsic property of a material, this is the geometric property of a material. The same material if you change its length or you change its area then its resistance will become different but its resistivity will remain the same. Resistivity only depends upon the temperature. As long as you are at standard temperature, the resistivity of two materials cannot be the same. But two materials can have the same resistance

So now, come to our equation. If I write here this equation that $$y \propto x$$ if $$c = 0$$ then $$y \propto x$$ and $$y = constant \times x$$ This implies $$y = mx$$So now I will have to write my equation in that form $$y \propto x$$, which means the dependent is proportional to the independent.

What is dependent and independent here? We give voltage and we get current.

This implies that the dependent variable is actually I and this is proportional to the independent.

Then this implies that this proportionality can be converted into an equation with a constant. What should be this constant? The equation is telling us that this constant is actually 1/R otherwise you cannot get the equation but how?

So, for this we will have to come back to our equation again means our plot again.

Let’ s say I discuss the very first response which is m = 1 which means that for one unit of V which is 1 volt, let’ s say one ampere current is produced.

Slope zero line means for any value of voltage you are not getting any current and undefined slope means for no value of current you are getting tremendous amount of current.

We know that Ohm’ s law is for ordinary conductors and ordinary conductors are that you give more voltage and you get less current.

m = 0 line is for ideal insulators. There are no free electrons, whatever voltage you give to this one, you will get the ideal insulator or a dielectric.

And when m = 1, you give 1 volt you get 1 ampere, the slope then shows ideal conductors or somehow superconductors.

We are in the ordinary conductors range, which is the area between the insulators and ideal conductors in the plot, that is where the Ohm’s law is valid.

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