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What's the difference between metal and normal high wattage resistors?

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That piece of wire is similar to the shunt you might find in a cheap digital multimeter. If so, it will typically be designed to drop 100 mV at 10 A so the power dissipated in it will be given by \$ P = VI = 0.1 \times 10 = 1 \ \text W \$. If it is any way decent it will have a tolerance of ±1% and be fairly stable with temperature.

The other is a 10 Ω wirewound resistor. It's resistance is too high to make a useful shunt, it will have inductance, might have a poor temperature coefficient and will probably have a tolerance of ±5% or ±10%.

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