Q93PE
Question
A short circuit in a\(120{\rm{ }}V\) appliance cord has a \(0.500{\rm{ }}\Omega \) resistance. Calculate the temperature rise of the \(2.00{\rm{ }}g\) of surrounding materials, assuming their specific heat capacity is and that it takes \(0.0500{\rm{ }}s\) for a circuit breaker to interrupt the current. Is this likely to be damaging?
Step-by-Step Solution
VerifiedThe rise in the temperature of a mass of surrounding materials is obtained as:
{{\Delta T}}\ =860^{o}C}
Damaging is likely to be possible.
Resistance is a measurement of the resistance to current flow in an electrical circuit. Resistance in ohms is denoted by the Greek letter omega\({\rm{(\Omega )}}\) .
The power of a process is the amount of some type of energy converted into a different type divided by the time interval \(\Delta t\) in which the process occurred:
\(P{\rm{ }} = {\rm{ }}\frac{{\Delta E}}{{\Delta t}}\)…………….(I)
If a potential difference \(\Delta V\) is maintained across a circuit element, the power, or the rate at which energy is supplied to the element, is:
\(P{\rm{ }} = {\rm{ }}I\Delta V\)
It is due to the potential difference across a resistor is given by \(\Delta V{\rm{ }} = {\rm{ }}IR\), we can express the power delivered to a resistor as:
\(\begin{aligned}{\underline{\phantom{xx}}}P{\rm{ }} = {\rm{ }}{I^2}R\\ = {\rm{ }}\frac{{{{(\Delta V)}^2}}}{R}\end{aligned}\)……………………(II)
The energy delivered to a resistor be electrical transmission appears in the form of internal energy in the resistor.
The energy of the value \(q\) is required to change the temperature of a mass \(m\) of a substance by an amount \(\Delta T\) is:
\(Q{\rm{ }} = {\rm{ }}mc\Delta T\)…………………(III)
The value of \(c\) is the specific heat of the substance.
Voltage across the appliance cord is: \(\Delta V{\rm{ }} = {\rm{ }}120{\rm{ }}V\).
Resistance of the short circuit is: \(R{\rm{ }} = {\rm{ }}0.500{\rm{ }}\Omega \).
Mass of the surrounding materials is: \(m{\rm{ }} = {\rm{ }}2.00{\rm{ }}g\).
Specific heat of the material is:
Time interval for the circuit breaker to interrupt the current is: \(\Delta t{\rm{ }} = {\rm{ }}0.0500{\rm{ }}s\).
Determiningthe temperature rise of a mass of surrounding materials.Also, determining if this is likely to be damaging.
Electrical energy supplied to the appliance cord is found in terms of the power \({\rm{P}}\) dissipated in it and the time interval \(\Delta t\) with the help of first equation as:
\(E{\rm{ }} = {\rm{ }}P\Delta t\)
Substituting the value of \(P\) with the help of second equation as:
\(E{\rm{ }} = {\rm{ }}\frac{{{{(\Delta V)}^2}}}{R}\Delta t\)
Putting the values and we get:
\(\begin{aligned}{c}E{\rm{ }} = {\rm{ }}\frac{{{{(120)}^2}(0.0500{\rm{ }}s)}}{{0.500{\rm{ }}\Omega }}\\ = {\rm{ }}1.44{\rm{ }} \times {\rm{ }}{10^3}{\rm{ }}J\end{aligned}\)
Assuming that all the electrical energy \(E\) is supplied to the cord is transferred into heat energy \(Q\) to the surrounding materials: \(E{\rm{ }} = {\rm{ }}Q\)
Substituting for the value of \(Q\) with the help of third equation as:
\(E{\rm{ }} = {\rm{ }}mc\Delta T\)
Solving for the value of\(\Delta T\):
\(\Delta T{\rm{ }} = {\rm{ }}\frac{E}{{mc}}\)
Putting the numerical values and we get:
Yes, this is likely damaging since the temperature difference is so great that it can melt the materials around.
Therefore, temperature rise of a mass of surrounding materials is: .
Yes, it is damaging.