The inner conductor of a coaxial cable has a radius of 0.900 mm, and the outer conductor's inside radius is 5.00 mm. The space between the conductors is filled with polyethylene, which has a dielectric constant of 2.30 and a dielectric strength of 18E6 V/m. What is the maximum potential difference that this cable can withstand?
no idea what to do...
thought it was
V= 18E6*d/k
but it was wrong...
18*10^6 V/m = 18*10^3 Volts. millimeter
Voltage = integral E dr from inner to outer r
2 pi E r = c constant (Gauss, therefore E is maximum beside the inner radius)
so
E = c/(2 pi r)
V = integral (c/2pi) dr/r
V = (c/2pi) ln(Router/Rinner)
V = (c/2pi) ln (5/.9) = 1.71 c/2 pi
now Emax is at r = .9
Emax = c/(.9*2pi)
c = .9*2pi (18*10^6)
so
V = 1.71 * .9 * 18 * 10^3
To find the maximum potential difference that the cable can withstand, we need to consider the electric field and breakdown voltage in the dielectric material.
First, let's calculate the electric field strength within the dielectric material. The electric field strength (E) inside a coaxial cable with a dielectric material is given by:
E = V / d,
where V is the potential difference across the cable, and d is the distance between the inner and outer conductors.
In this case, the distance between the conductors is the difference between their radii: d = (5.00 mm - 0.900 mm) = 4.10 mm = 0.00410 m.
So, we have:
E = V / 0.00410,
Next, we need to consider the dielectric strength of the polyethylene material. The dielectric strength (E_breakdown) is the maximum electric field strength that the material can withstand before it breaks down.
Given that the dielectric strength of polyethylene is 18E6 V/m, we can set up an equation:
E_breakdown = 18E6 V/m.
Now, we can solve for the maximum potential difference (V_max) that the cable can withstand. Rearranging the equation, we have:
V_max = E_breakdown * d.
Plugging in the known values:
V_max = (18E6 V/m) * 0.00410 m = 73,800 V.
Therefore, the maximum potential difference that this cable can withstand is 73,800 volts.
To find the maximum potential difference that the cable can withstand, we need to calculate the maximum electric field strength that the dielectric material can handle.
The electric field strength inside the dielectric material can be calculated using the formula: E = V/d, where V is the potential difference and d is the distance between the conductors.
In this case, the distance between the conductors is the difference between the outer and inner radii of the coaxial cable.
d = (outer radius) - (inner radius)
= 5.00 mm - 0.900 mm
= 4.10 mm
= 4.10 x 10^-3 m
Now, we can plug this value, along with the given dielectric constant, into the formula for the electric field strength:
E = V/d
Rearranging the formula, we get:
V = E x d
The dielectric strength of polyethylene is given as 18E6 V/m. So, we can substitute this value for E and the distance, d, to find the potential difference, V.
V = (18E6 V/m) x (4.10 x 10^-3 m)
V = 73.8 kV
Therefore, the maximum potential difference that this cable can withstand is 73.8 kilovolts (kV).