Consider a coaxial cable with a central conducting wire (r=1mm) surrounded by an outer coaxial conducting sheath (rinner=3mm,router=3.2mm) as shown. A current i=300mA flows down the center wire (into the page) and back through the conducting sheath (out of the page).
#4A (2 points possible)
Where does the resulting magnetic field have the greatest magnitude?
- unanswered At the center of the inner wire. At the surface of the inner wire. At the inner surface of the sheath. At the outer surface of the sheath.
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#4B (4 points possible)
What is the magnitude of the maximum magnetic field in T?
What is the magnitude of the maximum magnetic field in T? - unanswered
To determine where the resulting magnetic field has the greatest magnitude, we can use the right-hand rule. According to the right-hand rule, the magnetic field forms concentric circles around a current-carrying wire, with the direction of the magnetic field determined by the direction of the current.
In this case, the current flows down the center wire and back through the conducting sheath. The magnetic field lines will therefore curl in a clockwise direction around the current flow.
The magnitude of the magnetic field is strongest when we are closest to the current-carrying wire. So, at the center of the inner wire, the field lines are concentrated and close together, resulting in the greatest magnitude of magnetic field.
Therefore, the correct answer is: At the center of the inner wire.
Now, let's move on to the second question.
To calculate the magnitude of the maximum magnetic field, we can use Ampere's law. Ampere's law states that the line integral of the magnetic field around a closed loop is proportional to the current passing through the loop.
In this case, the cylindrical shape of the coaxial cable makes it convenient to use a circular Amperian loop. We can choose a circular loop of radius r within the coaxial cable, such that it is coaxial with the central wire.
The magnetic field produced by the current in the central wire will be parallel to the circular loop, and its magnitude will be the same at all points of the loop. Therefore, we can pull it out of the integral sign.
Integrating the magnetic field along the circular loop, we get:
B * 2πr = μ0 * Ienclosed,
where B is the magnetic field, r is the radius of the loop, μ0 is the permeability of free space, and Ienclosed is the current enclosed by the loop.
In this case, the current enclosed by the loop is the same as the current flowing through the central wire, which is given as 300mA or 0.3A.
Let's plug in the values:
B * 2πr = (4π * 10^-7 T·m/A) * 0.3A,
B * 2πr = 1.2π * 10^-7 T·m,
B = 0.6 * 10^-7 T.
Therefore, the magnitude of the maximum magnetic field is 0.6 * 10^-7 Tesla (T).