A conducting rod of mass 30 g and length 48 cm has holes drilled near each end so that it is free to slide in a vertical direction along two conducting rails as shown in the figure below. A current of 240 A is made to flow through an identical rod at the bottom of the assembly and return through the sliding sliding rod. (Notice the placement of an insulator.) Determine the height of the sliding rod above the base rod when in equilibrium. Assume the horizontal rods are very long.
________cm?
To determine the height of the sliding rod above the base rod when in equilibrium, we can analyze the forces acting on the system.
Step 1: Calculate the weight of the sliding rod.
The weight (W) can be calculated using the formula: W = mg, where m is the mass of the rod and g is the acceleration due to gravity.
Given:
Mass of the rod (m) = 30 g = 0.03 kg
Acceleration due to gravity (g) = 9.8 m/s^2
W = 0.03 kg * 9.8 m/s^2
W = 0.294 N
Step 2: Determine the force exerted on the sliding rod by the magnetic field.
The force exerted on a current-carrying conductor in a magnetic field can be found using the formula: F = BIL, where B is the magnetic field strength, I is the current, and L is the length of the conductor.
Given:
Current (I) = 240 A
Length of the rod (L) = 48 cm = 0.48 m
F = BIL
Step 3: Calculate the magnetic field strength.
Given that an identical rod at the bottom of the assembly has a current of 240 A, we can use the force equation to find the magnetic field strength (B).
F = BIL
Rearranging the formula:
B = F / (IL)
B = 0.294 N / (240 A * 0.48 m)
B ≈ 0.00205 T
Step 4: Find the force acting on the sliding rod.
Using the magnetic field strength (B) and the current (I) for the sliding rod, we can calculate the force (F) acting on the rod.
F = BIL
F = 0.00205 T * 240 A * 0.48 m
F ≈ 0.2352 N
Step 5: Determine the equilibrium height.
In equilibrium, the weight of the sliding rod is balanced by the force exerted by the magnetic field. So, the force acting on the sliding rod (F) must be equal to its weight (W).
F = W
0.2352 N = 0.294 N
Since the force and weight are equal, we can conclude that the sliding rod is in equilibrium.
Therefore, the height of the sliding rod above the base rod when in equilibrium is given as: 0 cm.
To determine the height of the sliding rod above the base rod when in equilibrium, we can use the concept of electromagnetic forces and the equation for mechanical equilibrium.
First, let's calculate the force acting on the sliding rod due to the magnetic field.
The magnetic force (F_mag) acting on a current-carrying conductor in a magnetic field with length (L), current (I), and magnetic field strength (B) can be calculated using the formula:
F_mag = BIL
where B is the magnetic field strength in Tesla, I is the current in Amperes, and L is the length of the conductor in meters.
In this case, the current flowing through the sliding rod and the bottom rod is the same (240 A). Since both rods are identical, the force acting on both will be the same.
Next, let's calculate the weight of the sliding rod.
The weight (W) of an object with mass (m) can be calculated using the formula:
W = mg
where m is the mass of the object in kilograms and g is the acceleration due to gravity (9.8 m/s^2).
In this case, the mass of the sliding rod is given as 30 g. We need to convert it to kilograms, so the mass becomes 0.03 kg.
Now, let's set up the equation for mechanical equilibrium.
When the sliding rod is in equilibrium, the magnetic force acting upwards on the sliding rod will be balanced by the weight of the sliding rod acting downwards. Therefore, we can write:
F_mag = W
Substituting the formulas for F_mag and W:
BIL = mg
Rearranging the equation to isolate the height (h):
h = (BIL) / (mg)
Note: Since the horizontal rods are assumed to be very long, the length of the conductor (L) will be the height (h) of the sliding rod.
We have all the required values, so let's substitute them into the equation:
h = (BIL) / (mg)
h = (B * 240 * 0.48) / (0.03 * 9.8)
Now, the final step is to calculate the value of h using the provided values of B (magnetic field strength) and 9.8 (acceleration due to gravity).