An open plastic soda bottle with an opening diameter of 2.5cm is placed on a table A uniform 1.75-7 magnetic field directed upward and oriented 25 degrees from vertical encompasses the bottle, what is the total magnetic flux through the plastic of the soda bottle&nbsp;

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An open plastic soda bottle with an opening diameter of 2.5cm is placed on a table A uniform 1.75-7 magnetic field directed upward and oriented 25 degrees from vertical encompasses the bottle, what is the total magnetic flux through the plastic of the soda bottle&amp;nbsp;

user_27qwe · Accepted Answer

To find the total magnetic flux through the plastic of the soda bottle, we need to calculate the magnetic flux passing through each small area element of the bottle&#039;s surface and then sum them up.The magnetic flux through a surface is given by the formula:Φ=∫𝐁·d𝐀where:- Φ is the magnetic flux,- 𝐁 is the magnetic field,- d𝐀 is the differential area vector.Since the magnetic field is uniform, we can consider it to be constant over the surface of the bottle. Therefore, we can simplify the integral to:Φ=𝐁·∫d𝐀Now, let&#039;s calculate the integral. The bottle&#039;s surface can be approximated as a cylinder with two circular bases and a curved side. We will divide the surface into three parts: the top circular base, the bottom circular base, and the curved side.First, let&#039;s calculate the magnetic flux through the top circular base. The area of a circle is given by:Atop=πrtop2where rtop is the radius of the circular top base. Since the opening diameter of the bottle is given as 2.5 cm, the radius can be calculated as:rtop=2.5cm2=1.25cm=0.0125mNow, let&#039;s calculate the magnetic flux through the top circular base:Φtop=𝐁·∫topd𝐀=𝐁·𝐀topΦtop=𝐁·πrtop2Next, let&#039;s calculate the magnetic flux through the bottom circular base. Since the magnetic field is directed upward and the surface is perpendicular to the field, the flux through the bottom base will be equal to the flux through the top base:Φbottom=Φtop=𝐁·πrtop2Finally, let&#039;s calculate the magnetic flux through the curved side. The curved side can be approximated as a rectangle with a width equal to the circumference of the circular base and a height equal to the height of the bottle. The circumference of a circle is given by:C=2πrtopThe height of the bottle is not provided in the question, so we&#039;ll assume it to be 1 meter for the sake of calculation.Height=1mNow, let&#039;s calculate the area of the curved side:Aside=C×Height=2πrtop×HeightΦside=𝐁·∫sided𝐀=B&amp;lt;br&amp;gt;·𝐀sideΦside=𝐁·(2πrtop×Height)Now, let&#039;s calculate the total magnetic flux through the plastic of the soda bottle by summing up the flux through each part:Φtotal=Φtop+Φbottom+ΦsideΦtotal=𝐁·πrtop2+𝐁·πrtop2+𝐁·(2πrtop×Height)Φtotal=2𝐁·πrtop2+𝐁·(2πrtop×Height)Substituting the given values, with the magnetic field strength of 1.75×10−7 T and the height of 1 meter, we get:Φtotal=2×(1.75×10−7)·π·(0.0125)2+(1.75×10−7)·(2π·0.0125·1)Calculating this expression will give us the total magnetic flux through the plastic of the soda bottle.

An open plastic soda bottle with an opening

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