A magnetic field is located within a region enclosed by an elliptical island with semi-minor axis of a=100m and semi-major axis of b=200m. A car carrying charge +Q=1.5C drives on the boundary of the island at a constant speed of v=5m/s and has mass m=2000kg. Any dimensions of the car can be assumed to be much smaller than the dimensions of the island. Ignore any contributions to the magnetic field from the moving car and assume that the car has enough traction to continue driving in its elliptical path.
Let the center of the island be located at the point (0,0) while the semi major and semi minor axes lie on the x and y-axes, respectively.
On this island, the magnetic field varies as a function of x and y:B(x,y)=kbecbxyz^ (pointing in the upward direction, perpendicular to the island plane in the positive z-direction). The constant cb=10−4m−2 and the constant kb=2.1μT
At what point on the island is the force from the magnetic field a maximum? Write the distance of this point from the x-axis in metres.
Given constants:
- Proton mass, mp=1.67⋅10−27kg
- Neutron mass, mn=1.67⋅10−27kg
- Electron mass, me=9.11⋅10−31kg
- Avogadro's constant, N0=6.02⋅1023mol−1
- Universal gas constant, R=8.31J/(mol⋅K)
- Boltzmann's constant, kB=1.38⋅10−23J/K
- Electron charge magnitude, e=1.60⋅10−19C
- 1 electron volt, 1eV=1.60⋅10−19J
- Speed of light, c=3.00⋅108m/s
- Universal Gravitational constant,
G=6.67⋅10−11(N⋅m2)/kg2
- Solar Mass
M⊙=1.988⋅1030kg
- Acceleration due to gravity, g=9.8m/s2
- 1 unified atomic mass unit,
1u=1.66⋅10−27kg=931MeV/c2
- Planck's constant,
h=6.63⋅10−34J⋅s=4.41⋅10−15eV⋅s
- Permittivity of free space,
ϵ0=8.85⋅10−12C2/(N⋅m2)
- Coulomb's law constant,
k=4πϵ01=8.99⋅109(N⋅m2)/C2
- Permeability of free space,
μ0=4π⋅10−7T⋅m/A
- Magnetic constant,
4πμ0=1⋅10−7(T⋅m)/A
- 1 atmospheric pressure,
1atm=1.01⋅105N/m2=1.01⋅105Pa
- Wien's displacement constant, b=2.9. 10−3m⋅K
- Stefan-Boltzmann constant,
σ=5.67⋅10−8W/m2/K4