Question

As a safety engineer, you must evaluate the practice of storing flammable conducting liquids in non-conducting containers. The company supplying a certain liquid has been using a squat, cylindrical plastic container of radius r = 0.20 mand filling it to height h = 10 cmwhich is not the container’s full interior height (figure). Your investigation reveals that during handling at the company, the exterior surface of the container commonly acquires a negative charge density of magnitude$\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\mu C}\mathbf{/}{\mathbf{m}}^{\mathbf{2}}$(approximately uniform). Because the liquid is a conducting material, the charge on the container induces charge separation within the liquid. (a) How much negative charge is induced in the centre of the liquid’s bulk?(b)Assume the capacitance of the central portion of the liquid relative to ground is 35 pFWhat is the potential energy associated with the negative charge in that effective capacitor?(c) If a spark occurs between the ground and the central portion of the liquid (through the venting port),the potential energy can be fed into the spark. The minimum spark energy needed to ignite the liquid is 10 mJ. In this situation, can a spark ignite the liquid?

Short answer

a) Amount of induced negative charge in the center of liquid bulk is$-0.50\mathrm{\mu C}$

b) Potential energy associated with the negative charge in effective capacitor is$3.6\times {10}^{-3}\mathrm{J}$

c) It will not cause a spark.

Step-by-step solution

The given data

a) Radius of cylinder, r = 0.20 m

b) Height of cylinder, h = 10 cm

c) Surface charge density,$\sigma =2.0\mathrm{\mu C}/{\mathrm{m}}^2$

d) The capacitance of the central portion of the liquid is C=35 pF

Understanding the concept of the capacitance

The ability of an electric conductor, or group of conductors, to hold separated electric charge per unit change in electrical potential is known as capacitance. We use the concept of surface charge density to find the charge in the center of liquid bulk. And using energy stored in the capacitor formula, we can find the potential energy associated with it.

Formulae:

The energy stored between the capacitors,$U=\frac{q^2}{2C}$ …(i)

The lateral surface area and base area of cylinder, $A=2\pi rh+\pi r^2$ …(ii)

The surface charge density of a body, $\sigma =\frac{q}{A}$ …(iii)

(a) Calculation of the amount of induced negative charge in the center of the liquid bulk

We can calculate the charged surface area of the cylindrical volume using equation (ii) as follows:

$$\begin{gathered} \mathrm{A}=2\mathrm{\pi }\left(0.20\mathrm{m}\right)\left(0.10\mathrm{m}\right)+\mathrm{\pi }{\left(0.20\mathrm{m}\right)}^2 \\ =0.25{\mathrm{m}}^2 \end{gathered}$$

From the figure, we see that only the bottom surface is charged not the top surface.

Now, the charge on the surface area is given using equation (iii) as follows:

$$\begin{gathered} \mathrm{q}=\mathrm{\sigma A} \\ =2.0\frac{\mathrm{\mu C}}{{\mathrm{m}}^2}\times 0.25{\mathrm{m}}^2 \\ =0.50\mathrm{\mu C} \end{gathered}$$

This is the charge induced on the exterior surface and according to the assumption in the problem same amount of charge is induced in the interior of the fluid.

Hence, the amount of induced negative charge in the center of the bulk is $-0.50\mathrm{\mu C}$.

(b) Calculation of the energy stored in the capacitor 

The energy stored in the capacitor is given by using the given data in equation (i) as follows:

$$\begin{gathered} U=\frac{{\left(5.0\times {10}^{-7}\mathrm{C}\right)}^2}{2\left(35\times {10}^{-12}\mathrm{F}\right)} \\ =3.6\times {10}^{-3}\mathrm{J} \end{gathered}$$

Hence, the value of the stored energy is $3.6\times {10}^{-3}\mathrm{J}$.

(c) Calculation of the reason if the liquid will ignite the spark or not

As given, the minimum energy required to cause a spark is 10 mJ .But the energy stored here is 3.6 mJ.

As the energy stored is less than the minimum, thus the energy required will not cause a spark.