# 10.6 Household wiring and electrical safety  (Page 4/11)

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## Summary

• The two types of electric hazards are thermal (excessive power) and shock (current through a person). Electrical safety systems and devices are employed to prevent thermal and shock hazards.
• Shock severity is determined by current, path, duration, and ac frequency.
• Circuit breakers and fuses interrupt excessive currents to prevent thermal hazards.
• The three-wire system guards against thermal and shock hazards, utilizing live/hot, neutral, and ground wires, and grounding the neutral wire and case of the appliance.
• A ground fault circuit interrupter (GFCI) prevents shock by detecting the loss of current to unintentional paths.

## Key equations

 Terminal voltage of a single voltage source ${V}_{\text{terminal}}=\epsilon -I{r}_{\text{eq}}$ Equivalent resistance of a series circuit ${R}_{\text{eq}}={R}_{1}+{R}_{2}+{R}_{3}+\text{⋯}+{R}_{N-1}+{R}_{N}=\sum _{i=1}^{N}{R}_{i}$ Equivalent resistance of a parallel circuit ${R}_{\text{eq}}={\left(\frac{1}{{R}_{1}}+\frac{1}{R2}+\text{⋯}+\frac{1}{{R}_{N}}\right)}^{-1}={\left(\sum _{i=1}^{N}\frac{1}{{R}_{i}}\right)}^{-1}$ Junction rule $\sum {I}_{\text{in}}=\sum {I}_{\text{out}}$ Loop rule $\sum V=0$ Terminal voltage of N voltage sources in series ${V}_{\text{terminal}}=\sum _{i=1}^{N}{\epsilon }_{i}-I\sum _{i=1}^{N}{r}_{i}=\sum _{i=1}^{N}{\epsilon }_{i}-I{r}_{\text{eq}}$ Terminal voltage of N voltage sources in parallel ${V}_{\text{terminal}}=\epsilon -I{\sum _{i=1}^{N}\left(\frac{1}{{r}_{i}}\right)}^{-1}=\epsilon -I{r}_{\text{eq}}$ Charge on a charging capacitor $q\left(t\right)=C\epsilon \left(1-{e}^{-\frac{t}{RC}}\right)=Q\left(1-{e}^{-\frac{t}{\tau }}\right)$ Time constant $\tau =RC$ Current during charging of a capacitor $I=\frac{\epsilon }{R}{e}^{-\frac{t}{RC}}={I}_{o}{e}^{-\frac{t}{RC}}$ Charge on a discharging capacitor $q\left(t\right)=Q{e}^{-\frac{t}{\tau }}$ Current during discharging of a capacitor $I\left(t\right)=-\frac{Q}{RC}{e}^{-\frac{t}{\tau }}$

## Conceptual questions

Why isn’t a short circuit necessarily a shock hazard?

We are often advised to not flick electric switches with wet hands, dry your hand first. We are also advised to never throw water on an electric fire. Why?

Not only might water drip into the switch and cause a shock, but also the resistance of your body is lower when you are wet.

## Problems

(a) How much power is dissipated in a short circuit of 240-V ac through a resistance of $0.250\phantom{\rule{0.2em}{0ex}}\text{Ω}$ ? (b) What current flows?

What voltage is involved in a 1.44-kW short circuit through a $0.100\text{-}\text{Ω}$ resistance?

12.0 V

Find the current through a person and identify the likely effect on her if she touches a 120-V ac source: (a) if she is standing on a rubber mat and offers a total resistance of $300\phantom{\rule{0.2em}{0ex}}\text{k}\text{Ω}$ ; (b) if she is standing barefoot on wet grass and has a resistance of only $4000\phantom{\rule{0.2em}{0ex}}\text{k}\text{Ω}$ .

While taking a bath, a person touches the metal case of a radio. The path through the person to the drainpipe and ground has a resistance of $4000\phantom{\rule{0.2em}{0ex}}\text{Ω}$ . What is the smallest voltage on the case of the radio that could cause ventricular fibrillation?

400 V

A man foolishly tries to fish a burning piece of bread from a toaster with a metal butter knife and comes into contact with 120-V ac. He does not even feel it since, luckily, he is wearing rubber-soled shoes. What is the minimum resistance of the path the current follows through the person?

(a) During surgery, a current as small as $20.0\phantom{\rule{0.2em}{0ex}}\mu \text{A}$ applied directly to the heart may cause ventricular fibrillation. If the resistance of the exposed heart is $300\phantom{\rule{0.2em}{0ex}}\text{Ω},$ what is the smallest voltage that poses this danger? (b) Does your answer imply that special electrical safety precautions are needed?

a. 6.00 mV; b. It would not be necessary to take extra precautions regarding the power coming from the wall. However, it is possible to generate voltages of approximately this value from static charge built up on gloves, for instance, so some precautions are necessary.

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