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
The greatest magnitude of the magnetic field will be at the center of the coaxial cable, between the inner and outer conducting sheaths.
Well, I would say the resulting magnetic field has the greatest magnitude in the middle of a Justin Bieber concert. Those screams can generate quite the magnetic field! But if we're talking about the coaxial cable, the magnetic field would likely have the greatest magnitude closer to the central conducting wire, since the current is flowing through it. Just like how the paparazzi gathers around Justin, the magnetic field would gather around the current-carrying wire.
To determine where the resulting magnetic field has the greatest magnitude, we can make use of Ampere's law. Ampere's law states that the magnetic field around a closed loop is directly proportional to the current passing through the loop.
In this case, the coaxial cable can be considered as a circular loop formed by the conducting sheath. The current passing through the loop is the total current i=300mA.
According to Ampere's law, the magnetic field would be strongest at the region closest to the current-carrying wire. Therefore, the greatest magnitude of the magnetic field would be at a distance r=1mm from the central conducting wire.
Hence, the resulting magnetic field has the greatest magnitude at a distance r=1mm from the central conducting wire.
To determine where the resulting magnetic field has the greatest magnitude, we can apply Ampere's law. Ampere's law states that the line integral of the magnetic field around a closed loop is equal to the product of the current enclosed by the loop and the permeability of free space (μ₀). Mathematically, it can be written as:
∮ B · dl = μ₀ * I_enclosed
In this case, we have a coaxial cable with a central conducting wire and an outer conducting sheath. The current i = 300mA flows down the central wire (into the page) and back through the conducting sheath (out of the page).
To find where the magnetic field has the greatest magnitude, let's consider a circular loop at a radial distance r from the central wire. The current enclosed by this loop is the sum of the current flowing through the central wire and the current flowing through the conducting sheath. Since the current flowing in opposite directions, the net current enclosed would be zero.
Therefore, the magnitude of the magnetic field will be zero for any radial distance r from the central wire. In other words, the resulting magnetic field has zero magnitude everywhere outside the central wire (r < 1mm) and the outer conducting sheath (r > 3.2mm).
Hence, the greatest magnitude of the resulting magnetic field will be at the inner surface of the outer conducting sheath (r = 3mm) where the current is concentrated.