Physics, 25.03.2020 00:35 andyboehm7411

You've made the finals of the Science Olympics! As one of your tasks, you're given 1.0 gg of aluminum and asked to make a wire, using all the aluminum, that will dissipate 6.5 WW when connected to a 2.5 VV battery.

a)- What diameter will you choose for your wire?

b)-What length will you choose for your wire?

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Answer from: lekab0001

The diameter is $D =1.319*10^{-7} m$

The length to choose is L= 3.57m

Explanation:

From the question we are told that

The mass of aluminum is $m_A = 1.0g = 1.0*10^{-3}kg$

The expected power to dissipate is P_e = 6.5W

The potential difference is E = 2.5V

First we need to obtain the resistance R which is mathematically represented as

$R = \frac{E^2}{P_e}$

Substituting values

$R = \frac{2.5^2 }{6.5}$

$= 0.962 \ \Omega$

Next we obtain the volume of the aluminium which is mathematically represented as

$v =\frac{m_A}{\rho_A}$

The value of the density of aluminium is $\rho_A=2700 kg/m^3$

Now substituting the value

$v = \frac{1.0*10^{-3}}{2700}$

$= 0.37 *10^{-6}m^3$

The resistance can also be mathematically represented as

$R = \frac{\rho L}{A}$

Where L is the length , A is the area and $\rho$ is resistivity

and for aluminum the resistivity is $\rho = 2.8*10^{-8} \Omega \ \cdot m$

Now multiplying the denominator and the numerator by L i.e $\frac{L}{L} = 1$

$R = \frac{\rho L}{A} [\frac{L}{L} ]$

$= \frac{\rho L ^2}{AL}$

But AL = v i.e area × length = volume

$R = \frac{\rho L ^2}{v}$

Now making L the subject of the formula in the question above

$L = \sqrt{\frac{Rv}{\rho} }$

$= \sqrt{\frac{(0.962)(0.37*10^{-6})}{2.8*10{-8} } }$

=3.57m

Volume can also be mathematically represented as

v =AL

Now making A the subject

$A = \frac{v}{L}$

$= \frac{0.37 *10 ^{-6}}{3.57}$

$=1.036*10^{-7}m^2$

The area A is mathematically represented as

$A = \pi r^2$

And $r = \frac{D}{2}$

Now substituting this into the formula for area

$A= \pi \frac{D^2}{4}$

Making D(diameter) the subject of the formula

$D = \sqrt{\frac{4A}{\pi} }$

$=\sqrt{\frac{4 * 1.036 *10^{-7}}{3.142} }$

$=1.319*10^{-7}m$

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You've made the finals of the Science Olympics! As one of your tasks, you're given 1.0 gg of aluminu...

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