a car is traveling at 40km/hr in a point of the region where the horizontal component of the earth’s magnetic field has the magnitude 2x10^-5 T, find the emf induced in the car’s vertical radio antenna which is 1.2m long.
To find the EMF induced in the car's vertical radio antenna, we can use Faraday's law of electromagnetic induction.
First, we need to find the magnetic flux through the antenna due to the Earth's magnetic field. The magnetic flux through a surface is given by the product of the magnetic field perpendicular to the surface and the area of the surface.
In this case, the area of the surface is equal to the length of the antenna multiplied by the perpendicular distance between the antenna and the Earth's magnetic field. Since the antenna is vertical, the perpendicular distance is equal to the length of the antenna itself, which is 1.2 m.
The magnetic flux through the antenna can be calculated as follows:
Flux = Magnetic Field x Area
Flux = (2x10^-5 T) x (1.2 m)
Next, we need to find the rate of change of magnetic flux. The car is moving horizontally, so the area of the surface is changing with time as the car moves. The rate of change of magnetic flux is given by the product of the rate of change of the area and the magnetic field.
The rate of change of magnetic flux can be calculated as follows:
Rate of change of flux = (2x10^-5 T) x (40 km/hr)
Now, we need to convert the units of the rate of change of flux to SI units:
Rate of change of flux = (2x10^-5 T) x (40,000 m/3600 s)
Rate of change of flux = 2.22x10^-4 T/s
Finally, we can use Faraday's law to find the induced EMF in the antenna. Faraday's law states that the induced EMF is equal to the negative rate of change of flux:
EMF induced = - Rate of change of flux
EMF induced = - 2.22x10^-4 T/s
The induced EMF in the car's vertical radio antenna is approximately -2.22x10^-4 T/s.
To find the induced EMF (electromotive force) in the car's vertical radio antenna, we can use Faraday's law of electromagnetic induction. According to this law, the induced EMF is equal to the rate of change of magnetic flux through the area enclosed by the antenna.
Let's break down the steps to find the induced EMF:
Step 1: Find the magnetic field strength (B):
Given in the question, the horizontal component of the Earth's magnetic field has a magnitude of 2x10^-5 T. Therefore, the magnetic field strength (B) in this scenario will also be 2x10^-5 T.
Step 2: Find the velocity (v):
The car is traveling at a speed of 40 km/hr. To use this value in the calculation, we need to convert it to meters per second (m/s). There are 1000 meters in a kilometer and 3600 seconds in an hour, so we can convert 40 km/hr to:
v = (40 km/hr) x (1000 m/km) / (3600 s/hr) = 11.11 m/s (rounded to two decimal places)
Step 3: Find the area enclosed by the antenna (A):
The length of the car's vertical radio antenna is given as 1.2 m. Since the area is a rectangle, the width can be considered negligible compared to the length, so the area can be approximated as the length of the antenna multiplied by some small value. Let's assume the width is 0.1 m:
A = (1.2 m) x (0.1 m) = 0.12 m^2
Step 4: Calculate the rate of change of magnetic flux (dΦ/dt):
The rate of change of magnetic flux is given by the equation:
dΦ/dt = B * A * v
Plugging in the values we found earlier:
dΦ/dt = (2x10^-5 T) * (0.12 m^2) * (11.11 m/s)
Calculating this equation will give us the rate of change of magnetic flux.
Step 5: Calculate the induced EMF:
Finally, we can substitute the value of the rate of change of magnetic flux into the formula for induced EMF, which is:
EMF = -dΦ/dt
Substitute the calculated value of dΦ/dt into this equation to find the induced EMF in the car's vertical radio antenna. Note that the negative sign represents the direction of the induced current.
By following these steps, you should be able to calculate the induced EMF in the car's vertical radio antenna.