Question

An open plastic soda bottle with an opening diameter of $\mathbf{2}\mathbf{.}\mathbf{5}\mathbf{}\mathbf{cm}$is placed on a table. A uniform $\mathbf{1}\mathbf{.}\mathbf{75}\mathbf{-}\mathbf{T}$magnetic field directed upward and oriented$\mathbf{25}\mathbf{°}$ from the vertical encompasses the bottle. What is the total magnetic flux through the plastic of the soda bottle?

Short answer

The total magnetic flux through the plastic of the soda bottle is$-7.79\times {10}^{-4}\mathrm{Wb}$

Step-by-step solution

Definition of magnetic flux

The term magnetic flux may be defined as the measurement of total magnetic field passing through the given area.

Determine the total magnetic flux through the plastic bottle

The sum of magnetic flux through the plastic and through the open cap is net magnetic flux through the bottle.

$$\begin{gathered} {\mathrm{f}}_{\mathrm{net}}={\mathrm{f}}_{\mathrm{plastic}}+{\mathrm{f}}_{\mathrm{cap}} \\ {\mathrm{f}}_{\mathrm{net}}=\mathrm{o} \\ {\mathrm{f}}_{\mathrm{plastic}}+{\mathrm{f}}_{\mathrm{cap}}=\mathrm{o} \\ {\mathrm{f}}_{\mathrm{plastic}}=-{\mathrm{f}}_{\mathrm{cap}} \end{gathered}$$

Now

$$\begin{gathered} {\mathrm{f}}_{\mathrm{plastic}}=-\mathrm{BAcos\theta } \\ {\mathrm{f}}_{\mathrm{plastic}}=-\mathrm{B}\left({\mathrm{\pi r}}^2\right)\mathrm{cos\theta } \\ {\mathrm{f}}_{\mathrm{plastic}}=-{\mathrm{\pi r}}^2\mathrm{Bcos\theta } \end{gathered}$$

Here $\mathrm{B},\mathrm{A},\mathrm{r}$and $\theta$are magnetic field, area of open cap, radius and angle between magnetic field and the area vector respectively.

Put all the given values and$\mathrm{r}=\mathrm{d}/2$

$$\begin{gathered} {\mathrm{f}}_{\mathrm{plastic}}=-\mathrm{\pi }{\left(1.25\times {10}^{-2}\mathrm{m}\right)}^2\left(1.75\mathrm{T}\right)\cos \left(25°\right) \\ {\mathrm{f}}_{\mathrm{plastic}}=-7.79\times {10}^{-4}\mathrm{Wb} \end{gathered}$$

Hence, the total magnetic flux through the plastic of the soda bottle is $-7.79\times {10}^{-4}\mathrm{Wb}$