# Simple ohms law on a battery ? Paradox or conceptual error?

Suppose we have a regular pencil battery which supplies DC voltage $V$. Say we take copper wire and connect the ends of the battery to an $R$ ohms resistance.

Then Ohm's law tells use the current in the wire is $\frac{V}{R}$.

This means as we keep decreasing the value of $R$, we will keep getting higher and higher values of current, since $V$ is fixed.

Now if we simply connect the ends of the battery by a copper wire without an intermediate resistance, of course the value of the current will not be infinity, but it will be $\frac{V}{R_{copper}}$ which is still very large.

The resistance of copper is so small that even for $V=1.5$ volts we will get current of larger than 1 Amp with copper. And according to this link 1 Amp can almost give you a heart attack.

So why is it that we dont hear about major accidents about people connecting two ends of a pencil battery with regular copper wire? Is there some fallacy in my reasoning above?

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## 2 Answers

An ampere passing through your heart can give you a heart attack. An ampere passing through a wire will not.

The human body has a fairly large resistance ($10000\ \mathrm{\Omega}$ perhaps?), so the same voltage that can make a large current pass through a copper wire will not necessarily make any significant current flow through a person.

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Thank you. Can we also say that if the battery has some kind of "inside" resistance R_in, the overall current in the battery is bounded by V/R_in and that for pencil batteries this maximum current turns out to be non-lethal? –  smilingbuddha May 12 at 17:58
@smilingbuddha, there is some internal resistance to any battery, and that will limit the total current the battery outputs exactly as you say. But that maximum current in general will be potentially lethal. And while you can measure the internal resistance of a battery, it is more difficult that just connecting it up to a single resistor. –  Colin McFaul May 12 at 18:46
Just came to say minor issue David: Though the average resistance of human body is around $10^4\Omega$, it's actually a variable resistor (i.e) its resistance is different at different parts of the body. In case of heart, several milliamperes is enough for our heart to shut down. More on hyper textbook –  Waffle's Crazy Peanut May 13 at 2:41
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First, one additional point. It's not just $R_{copper}$ limiting current, but also the battery's internal resistance. This is modeled as one more resistor in series. For most small batteries you might get your hands on, it's less than one Ohm.

People playing with wire and low-voltage batteries don't get zapped because the resistance of Human, when contacted on more or less normal skin, is thousands to hundreds of thousands of Ohms.

For a contrasting point of view, one may try connecting the battery with wires (any metal) to one's tongue. (Best done as a thought experiment...)

When medics start a patient's heart with a zapper, they're using high voltages, hundreds to thousands of volts, to overcome skin contact resistance. On the other hand, contact with moist internal tissues means low resistance, dozens of Ohms probably, and small currents are enough to have interesting consequences. Read about pacemakers for more.

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