Voltage drop across battery with internal resistance

Consider a battery with internal resistance connected to an external voltage source; there is a voltage difference along the battery The voltage source is not drawn in the picture

The voltage difference is $\ V= E ± Ir$ where ± is determined by the direction of the current flow. The symbols have their usual meanings V is the total voltage difference through the battery. As per the definition of resistance ; the resistance of some component is equal to the total voltage difference through that component divided by the total current that flows through it.

Accordingly, it would follow that $\frac{V}{r}=I$ Which is in contradiction to the aforementioned. E would be zero, which is clearly wrong

The second method implicitly assumes that the total voltage drop is only caused by the resistance alone. However to my knowledge resistance is defined as the ratio between the total voltage difference and the current that flows through it.

What is the correct definition of resistance of a component, if it is different from the above? Or is there something wrong in my reasoning?

Reading the answer I now consider the contexts where V=IR can be used as a definition for the resistance of a component. https://en.m.wikipedia.org/wiki/Electrical_resistance_and_conductance The wiki article uses “object” implying, some sort of a generality. $V$-$I$ linear OR $V=IR$ is statement of Ohm's law? Too stresses how resistance is defined.

• I haven't the foggiest idea of what you're trying to say. – Hot Licks Jul 6 '18 at 20:40

Where have I gone wrong?

You've gone wrong here:

the resistance of some component is equal to the total voltage difference through that component divided by the total current that flows through it.

It's not some component that Ohm's law applies to, it's specifically (ideal) resistors.

To model a physical cell, one starts by putting an ideal resistor in series with an ideal voltage source. You'll note that the voltage across the resistor is given by Ohm's law but the voltage across the voltage source is what it is regardless of the current through, i.e., the ideal voltage source does not obey Ohm's law.

Thus, the voltage across the series combination of the voltage source and resistor will not, and should not be expected to, obey Ohm's law.

Seeing the updates to your original post, I think I better see what you're asking about.

As I wrote above, Ohm's law $V = I R$ where $R$ is a constant applies to ideal resistors.

However, for non-ohmic components, one can define a static (or DC) resistance $R_{DC}$ as well as a dynamic (or small-signal) resistance $r$.

The static resistance is simply the ratio of the DC voltage across and current through:

$$R_{DC} \equiv \frac{V_{DC}}{I_{DC}}$$

So, in the battery example of your question, the static resistance is given by

$$R_{DC} = \frac{E + I_{DC}r}{I_{DC}} = \frac{E}{I_{DC}} + r$$

(Note: this is a somewhat unusual application of the static resistance concept since the battery is a typically a source rather than a load).

The The dynamic resistance is the ratio of the change in voltage to the change in current from their DC values:

$$r \equiv \frac{dV}{dI} \approx \frac{\Delta V}{\Delta I}$$

In your battery example, the dynamic resistance is just the internal resistance $r$

• what is the scope of the objects that you call ideal resistors; physics.stackexchange.com/questions/339844/… this suggests how this V=IR statement is applicable to a range of objects, as does the wiki article en.m.wikipedia.org/wiki/Electrical_resistance_and_conductance This makes me wonder whether this invalidity is limited to objects that cause a voltage difference itself; i.e. batteries – SNB Jul 6 '18 at 17:16
• @SNB, Ohm's law doesn't apply to: capacitors, inductors, current sources, voltage sources, switches, diodes, transistors, vacuum tubes, etc. etc. etc. There are, however, the notions of the static $(R_{DC} \equiv V_{DC}/I_{DC})$ and dynamic $(r_d \equiv \Delta V_d/\Delta I_d)$ resistance for devices that do not obey Ohm's law. I can address this in an addendum to my answer if you think it would be helpful. – Hal Hollis Jul 6 '18 at 19:54
• In your second statement concerning the definition of $\ R_{DC}$, can we associate some physical meaning to $\frac{E}{I_{DC}}$? Also, for ohmic substances would these two definitions become equivalent? – SNB Jul 7 '18 at 5:34
• A light bulb will have specifications like wattage, voltage etc Eg- 60W, 230V. We can calculate the resistance at operating temperature, from these. So, does static resistance correspond to this resistance, ( Siddharth’s answer in this quora.com/…, seems to suggest so), while dynamic resistance denotes an instantaneous value of resistance? I suppose the internal resistance of a battery would change with time. Is this why the dynamic resistance is the same as the internal resistance? – SNB Jul 7 '18 at 5:58
• @SNB, say that the bulb has 230V across and you measure the current to be 0.261A (rms values), then the static resistance is about 882 ohms. Now change the voltage to 231V and measure the current. The dynamic resistance is the change in voltage (1V) divided by the change in current (it is the slope of the V-I curve at V=230V). Note that for an ideal resistor only, the static and dynamic resistance are equal. – Hal Hollis Jul 7 '18 at 13:27