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When there is a battery connected in an open circuit, there is some induced e.m.f, right?

Now, if a battery is connected across its ends with a wire, with no resistances or any other appliances, and adding that the battery is perfectly ideal with no internal resistance, then still, there will be a current through the wire, right?

Then, what is the purpose of resistances? Don’t they offer obstruction to the flow of current and the battery must to work to propel an electron through it?

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If you connect the pins of your battery with a wire with negligible resistance a lot of current will flow through the wire, and the battery won't last long.

This is equivalent to having a dam and opening it completely, letting all the water flow out of the basin in no time.

Sometimes you need just a little current to power your electronic device and hence you need resistance to reduce the current flow out of your battery.

Following the example of the dam, if you open the dam completely you risk destroying everything that you find downstream, in the same way you use resistance to control the current flow across sensitive equipment.

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The reason why resistors are extremely necessary is because of the danger of humungous currents that a short circuited wire can create. In reality, no wire is resistanceless, there's bound to be some resistance in it, no matter how small. The power dissipated by a resistor is given by

$$P=I^2 R=VI$$

In a circuit with extremely low value of resistance, the curremt blows up to a huge amount (because $I=V/R$). But the voltage across the whole setup remains equal to the external applied voltage (which is generally around $5-10\:\rm V$ for school experiments). Thus the resulting power ($VI$) takes on a huge value, burning up the circuit (by melting the wire) and damaging every other device connected to that circuit. Not at all a good scenario.

Thus to control and limit the amount of current passing through a circuit, we use resistors.

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When there is a battery connected in an open circuit, there is some induced e.m.f, right?

Correct.

Now, if a battery is connected across its ends with a wire, with no resistances or any other appliances, and adding that the battery is perfectly ideal with no internal resistance, then still, there will be a current through the wire, right?

Correct again, but all wires have some resistance and all real batteries have some internal resistance.

Then, what is the purpose of resistances? Don’t they offer obstruction to the flow of current and the battery must to work to propel an electron through it?

In addition to the purposes cited in the other answers, resistors are use to intentionally dissipate large amounts of heat in electrical heating appliances. Typically, the resistor consists of a special metal alloy Nickel Chromium. Sometimes called a Nichrome wire.

Hope this helps.

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Let's start with a simple explanation of the quantities involved.

Charge: Basically the amount of charged particles (almost always electrons) inside a certain volume. It's measured in coulomb (C).

Tension: The energy needed to move 1 C of charge from point A to point B in an electric field (or circuit, most commonly). It's measured in volt (V), where 1 V = 1 J/C.

Current: The amount of charge that flows through a surface (usually a section of an electric wire) in a second. It's measured in ampere (A), where 1 A = 1 C/s.

Resistance: It's defined by Ohm's law as the ratio of tension and current: $R = \frac{U}{I}$. It's measured in ohm ($\Omega$), where $1\ \Omega = 1\ \frac{\mbox{V}}{\mbox{A}}$.

With that cleared, let's start with the first piece of the circuit. A battery is a device that stores a certain charge. To simplify it a lot, a battery is composed of two chemicals, one that wants to emit electrons, and one that wants to absorb them. The only way to obtain this effect is by connecting the two ends of the battery with a conductor.

Now, if nothing impedes the flow of electrons (so ignoring any form or resistance), the negative terminal will emit all of its electrons at the same time, and they will rapidly go to the positive terminal. In a tiny fraction of a second the battery becomes completely useless.

If we consider instead the effect of the tiny internal resistance of the battery and the tiny resistance of the wire, we have a really high current that goes through a resistor. And when a resistor is traversed by current, it dissipates energy by emitting heat. Remembering that the power (energy per second) emitted is $P = VI$ and using Ohm's law to get rid of the tension, we get $P = RI^2$. Since the current is very high, the power emitted is also considerable, and you can have quite a few issues with a short-circuit. Wires and batteries can melt, and I think some older models of batteries can even explode. I suspect Youtube has some videos about this topic, and they can be both instructive and entertaining, as long as you don't try them at home under any circumstance.

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what is the purpose of resistances? Don’t they [obstruct] the flow of current?

The purpose of a resistor is to be a thing that obeys Ohm's Law. That is to say, it's a thing that enforces a linear relationship between the current flowing through it and the voltage between its two ends.

Sometimes we use a resistor when we want the current in some circuit to vary in proportion to the voltage that is forced upon the resistor. Other times, we use it when we want the voltage to vary in proportion to the current that is forced through it. Ohm's Law works either way.

Yes. Resistors waste power. Just like how cars waste fuel*. In electrical engineering, as in most other forms of engineering, one always has to trade-off pluses against minuses. We usually try to find a design that wastes the least possible amount of power while still achieving other goals including monetary goals (cost of the design, cost of manufacturing, cost of operation.)


* In theory, in an ideal automobile, with regenerative braking, you should be able to leave your home, go anywhere you please, and return home with no net energy consumed. If any energy is consumed, that's because your automobile is not 100% efficient.

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The simplest answer is that, we need to adjust the amount of voltage that we actually want to give an electrical appliance. Pushing through much current through an appliance can damage it

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I work in an industrial environment. If we scale up your battery to 100 amps using 480 volts per leg of 3 phase wiring (which is common) then your short circuit at the disconnect is 48,000 amps. At this point the copper in the contacts vaporizes and explodes in an arc flash explosion.

You need the resistance to keep your current level at the point that is safe for your circuit. You do not want to supply 10 amps to a circuit that only needs 1 amp. This is why you use resisters. Interestingly, they can help increase current in a circuit by putting them in parallel. The formula for resisters in parallel is Rt = (1/R1 + 1/R2 + 1/Rn)

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