A square bar with a mass of 0.830 kg and 19.3cm long (l) rests on two parallel horizontal rails that are 18.0cm apart (d) and 120.0cm long (L) as shown in the figure. (By horizontal it is meant that the rails lie in a plane perpendicular to the local gravitational field.) The bar carries a current, provided by the rails, of I=5.3 mA in the direction shown causing the bar to slide along the rails without friction. A uniform magnetic field of magnitude |B| = 0.560T is directed perpendicular to the rod and the rails.
To find the magnetic force experienced by the square bar, we can use the equation:
F = I * L * B * sin(θ)
where:
- F is the magnetic force
- I is the current flowing through the bar
- L is the length of the bar that is perpendicular to the magnetic field
- B is the magnitude of the magnetic field
- θ is the angle between the magnetic field and the current direction
In this case, the square bar is sliding along the rails, so the length of the bar perpendicular to the magnetic field is its width (w).
We can calculate the width (w) of the square bar using the Pythagorean theorem:
w^2 = l^2 - d^2
where:
- w is the width of the square bar
- l is the length of the bar
- d is the distance between the rails
Substituting the given values:
w^2 = (0.193m)^2 - (0.18m)^2
Calculating the width of the square bar:
w^2 = 0.037249m^2
Taking the square root of both sides:
w = 0.193m
Now we can calculate the magnetic force (F) using the equation:
F = (5.3mA)(0.193m)(0.560T) * sin(90°)
Converting current to amperes:
I = 5.3mA = 5.3 × 10^(-3) A
Calculating the magnetic force:
F = (5.3 × 10^(-3) A)(0.193m)(0.560T) * sin(90°)
Finally, we can calculate the magnetic force. Let's plug in the values and calculate:
F = 5.3 × 10^(-3) A * 0.193m * 0.560T = 5.3 × 10^(-3) * 0.193 * 0.560 N = 0.057464 N
Therefore, the magnetic force experienced by the square bar is approximately 0.057464 N.
To find the magnetic force acting on the square bar, we can use the formula for the magnetic force on a current-carrying conductor in a magnetic field.
The formula is given by:
F = BIL
where:
F is the magnetic force,
B is the magnetic field strength,
I is the current, and
L is the length of the conductor.
In this case, we are given the values for B, I, and L, but we need to find the value of the current-carrying square bar's length.
Given that the square bar is 19.3 cm long, we can calculate its length using the Pythagorean theorem, as the square bar forms a right-angled triangle with the rails.
The formula for the length of the square bar (l) can be given as:
l = sqrt(d^2 + L^2)
where:
d is the distance between the rails (18.0 cm) and
L is the length of the rails (120.0 cm).
Let's calculate the length of the square bar (l):
l = sqrt((18.0 cm)^2 + (120.0 cm)^2)
l = sqrt(324 cm^2 + 14,400 cm^2)
l = sqrt(14,724 cm^2)
l = 121.386 cm
Now, we can substitute the values into the magnetic force formula to find the magnetic force (F):
F = (0.560 T)(5.3 mA)(121.386 cm)
However, before calculating further, we need to convert the milliamps (mA) to amperes (A) and the centimeters (cm) to meters (m) as the SI units for current and length are A and m, respectively.
1 A = 1000 mA (conversion factor for current)
1 m = 100 cm (conversion factor for length)
Converting the current:
I = 5.3 mA / 1000
I = 0.0053 A
Converting the length:
L = 121.386 cm / 100
L = 1.21386 m
Now, we can substitute the converted values into the formula to find the magnetic force (F):
F = (0.560 T)(0.0053 A)(1.21386 m)