Question

On its highest power setting, a microwave oven increases the temperature of 0.004 kgof spaghetti by $\mathbf{45}\mathbf{.}\mathbf{0}\mathbf{°}\mathbf{C}$ in 120 s. (a) What was the rate of power absorption by the spaghetti, given that its specific heat isrole="math" localid="1655872138905" $\mathbf{3}\mathbf{.}\mathbf{76}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{}\mathbf{J}\mathbf{/}\mathbf{kg}\mathbf{\times }\mathbf{°}\mathbf{C}$? (b) Find the average intensity of the microwaves, given that they are absorbed over a circular area 20.0 cmin diameter. (c) What is the peak electric field strength of the microwave?(d) What is its peak magnetic field strength?

Short answer

a. The rate of power absorption by the spaghetti is 564 W .

b. The average intensity of the microwaves is$1.80\times {10}^4\mathrm{W}/{\mathrm{m}}^2$

c. The peak electric field strength of the microwaves is$3.68\times {10}^4\mathrm{V}/\mathrm{m}$

d. The peak magnetic field strength of the microwaves is$1.23\times {10}^{-5}\mathrm{T}$

Step-by-step solution

Definition of Concept

A temperature is a unit of measurement for hotness or coldness that can be stated in a variety of ways, including Fahrenheit and Celsius. The direction in which heat energy will flow spontaneously is determined by temperature.

Findtherate of power absorption by the spaghetti

(a)

Considering the given information,

Mass: m=0.4 kg .

Specific heat is:$\mathrm{c}=3.76\times {10}^3\mathrm{J}/\mathrm{kg}.°\mathrm{C}$

The temperature increase is: $∆\mathrm{T}=45°\mathrm{C}$.

The Time interval is: $\mathrm{t}=120\mathrm{s}$.

Apply the formula,

The formula for calculating the rate of power absorption is:

$\mathrm{P}=\frac{\mathrm{mc}∆\mathrm{T}}{\mathrm{t}}$

Putting the values,

$$\begin{gathered} \mathrm{P}=\frac{\left(0.40\mathrm{kg}\right)\times \left(3.76\times {10}^3\mathrm{J}/\mathrm{kg}.°\mathrm{C}\right)}{120\mathrm{s}} \\ =564\mathrm{W} \end{gathered}$$

Therefore, the required rate of power absorption is 564 W .

Findtheaverage intensity of the microwaves

(b)

Considering the given information,

The P=564 W is the rate of power absorption and the diameter is d=20 cm

Apply the formula,

The formula for determining intensity is as follows:

$I=\frac{P}{\mathrm{\pi }{\displaystyle \frac{{\mathrm{d}}^2}{4}}}$

Putting the values,

$$\begin{gathered} \mathrm{I}=\frac{4\times 564}{3.14\times {\left(0.20\mathrm{m}\right)}^2} \\ =1.80\times {10}^4\mathrm{W}/{\mathrm{m}}^2 \end{gathered}$$

Therefore, the required intensity of the microwaves is $1.80\times {10}^4\mathrm{W}/{\mathrm{m}}^2$.

Findthepeak electric field strength of the microwaves

(c)

Considering the given information,

Microwaves have an intensity of $\mathrm{I}=1.80\times {10}^4\mathrm{W}/{\mathrm{m}}^2$.

Apply the formula,

The following is the relationship between intensity and electric field strength:

$\mathrm{I}=\frac{1}{2}{\mathrm{c\epsilon }}_0{\mathrm{E}}_0^2$

Microwaves have a peak electric field strength of,

$$\begin{gathered} {\mathrm{E}}_0=\sqrt{\frac{2\mathrm{I}}{{\mathrm{c\epsilon }}_0}} \\ =\sqrt{\frac{2\times \left(1.80\times {10}^4\mathrm{W}/{\mathrm{m}}^2\right)}{\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)\left(8.85\times {10}^{-12}{\mathrm{C}}^2/\mathrm{N}·{\mathrm{m}}^2\right)}} \\ =3.68\times {10}^3\mathrm{V}/\mathrm{m} \end{gathered}$$

Therefore, the required peak electric field strength of the microwaves is $3.68\times {10}^3\mathrm{V}/\mathrm{m}$.

Findthepeak magnetic field strength of the microwaves

(d)

Considering the given information,

Peak electric field strength is ${\mathrm{E}}_0=3.68\times {10}^3\mathrm{V}/\mathrm{m}$.

Apply the formula,

The magnetic field and electric field strength are related in the following way:

${\mathrm{B}}_0=\frac{{\mathrm{E}}_0}{\mathrm{c}}$

Putting the values,

$$\begin{gathered} {\mathrm{B}}_0=\frac{3.68\times {10}^3\mathrm{V}/\mathrm{m}}{3\times {10}^8\mathrm{m}/\mathrm{s}} \\ =1.23\times {10}^{-5}\mathrm{T} \end{gathered}$$

Therefore, the required peak magnetic field strength of the microwaves is $1.23\times {10}^{-5}\mathrm{T}$.