A uniform horizontal wire with a linear mass density of 0.48 g/m carries a 1.9 A current. It is placed in a constant magnetic field, with a strength of 3.5 10-3 T, that is horizontal and perpendicular to the wire. Neglect the magnetic field of the Earth.
(a) As the wire moves upward starting from rest, what is its acceleration?
m/s2
Net force F = F(Ampere) –m•g = I•L•B•sinα –d•L.
(sinα = 1, d =0.48•10^-3 kg/m)
F = m•a =d•L•a
I•L•B –d•L = d•L•a.
a =(I •B) /d – 1.
Why did the wire bring a ladder to the party? Because it wanted to amp up the fun!
Now, let's calculate the wire's acceleration. To do that, we can use the formula:
a = (μ₀ I B)/(ρ L),
Where:
a = acceleration,
μ₀ = magnetic constant (4π x 10^-7 T m/A),
I = current in the wire,
B = magnetic field strength,
ρ = linear mass density of the wire, and
L = length of the wire.
Plugging in the values, we get:
a = (4π x 10^-7 T m/A) * (1.9 A) * (3.5 x 10^-3 T) / (0.48 x 10^-3 kg/m * L).
However, we don't know the length of the wire, so we need more information to provide an exact answer.
To find the acceleration of the wire, we can use the magnetic force equation:
F = BIL
where:
F = magnetic force
B = magnetic field strength
I = current
L = length of the wire
In this case, the magnetic field is perpendicular to the wire and points horizontally, so the magnetic force will act vertically upward.
The equation for magnetic force can be rewritten as:
F = ma
where:
m = mass
a = acceleration
To find the acceleration, we need to determine the mass of the wire. We can do this by multiplying the linear mass density (μ) by the length of the wire (L):
m = μL
Given:
Linear mass density (μ) = 0.48 g/m
Length of the wire (L) = unknown
We can rearrange the equation to solve for the length of the wire:
L = m / μ
Substituting the values:
L = unknown / (0.48 g/m)
Since we don't have the length of the wire, we cannot directly calculate the acceleration.
To find the acceleration of the wire, we can use Newton's second law of motion, which states that the net force on an object is equal to its mass multiplied by its acceleration (F = ma).
First, let's find the force on the wire due to the magnetic field. The magnetic force on a current-carrying wire can be calculated using the equation:
F = ILBsinθ
where F is the magnetic force, I is the current, L is the length of the wire, B is the magnetic field strength, and θ is the angle between the direction of the current and the magnetic field.
In this case, the wire is moving upward, so the angle θ between the current and the magnetic field is 90 degrees, making sinθ equal to 1.
The length of the wire doesn't affect the force since the wire's mass density is already given. Therefore, the equation simplifies to:
F = IB
Now, let's substitute the given values:
I = 1.9 A (current)
B = 3.5 x 10^-3 T (magnetic field strength)
F = (1.9 A) * (3.5 x 10^-3 T)
Calculating the multiplication:
F = 6.65 x 10^-3 N
Now, using Newton's second law, we have:
F = ma
Substituting the force and rearranging the equation to solve for acceleration:
a = F / m
The mass (m) of the wire can be determined by finding the mass per unit length and multiplying it by the length of the wire traveled.
The linear mass density of the wire is given as 0.48 g/m. To convert this to kg/m, we divide by 1000:
linear mass density in kg/m = (0.48 g/m) / 1000 = 0.00048 kg/m
Now we can find the mass of the wire by multiplying the linear mass density by the length of the wire traveled, which is not given in the question:
m = (0.00048 kg/m) * L
Substituting this value into the equation for acceleration:
a = (6.65 x 10^-3 N) / (0.00048 kg/m) * L
Finally, we need the value of the length of the wire traveled (L) to find the acceleration. Unfortunately, the question does not provide this information. Without knowing the length, we cannot calculate the acceleration.