91Ó°ÊÓ

Electromagnetic induction is a fascinating phenomenon. It occurs when a conductor or coil experiences a change in magnetic field. Faraday's Law helps us quantify this interaction. It states that the induced electromotive force (EMF) in a loop is equal to the change in magnetic flux through the loop per unit time.

Mathematically, this can be expressed as: \[ \epsilon = - \frac{d \Phi}{dt} \] where:

• \( \epsilon \) is the induced EMF
• \( \Phi \) represents the magnetic flux
• \( \frac{d \Phi}{dt} \) is the rate of change of the flux

In the scenario with the tilted rail and rod, the EMF is generated because the rod moves relative to a uniform magnetic field. Symmetrical to Faraday's Law, the expression \( \epsilon = Blv\sin\theta \) applies here.

When the rod slides down the rails, it cuts through the magnetic field lines. This action induces an EMF since the system acts like a loop, where \( B \) is the field strength, \( l \) is the rod's length, \( v \) the velocity, and \( \theta \) the angle between the motion and the magnetic field (90° in this case). This directly connects how movement in a magnetic field can lead to current induction.

Magnetic forces come into play when the rod carries a current and moves within a magnetic field. This force is responsible for balancing the gravitational pull down the inclined rails in the exercise. Magnetic force \( F_B \) on a current-carrying conductor is given by the equation: \[ F_B = IlB\sin\theta \] where:

• \( I \) is the current flowing through the conductor
• \( l \) is the length of the conductor within the magnetic field
• \( B \) is the magnetic field strength
• \( \theta \) is the angle between the magnetic field and the direction of the current (90° here)

Here, we see that the angle \( \theta \) is crucial; the full force effect occurs when \( \theta \) equals 90 degrees, where \( \sin 90^{\circ} = 1 \). The condition simplifies our equation to \( F_B = IlB \), which is used in our solution.

This magnetic force is key to understanding how the system remains in equilibrium. It exactly counteracts the gravitational force acting on the rod down the slope. Therefore, the gravitational component \( mg\sin\alpha \) equates to \( IlB \). Solving this system tells us the current \( I \) necessary to maintain the rod's constant velocity down the ramps.

Newton's Second Law is a cornerstone of physics. It states that the force acting upon an object is equal to the mass of that object multiplied by its acceleration. In formula terms: \[ F = ma \] where:

• \( F \) is the force
• \( m \) is the mass
• \( a \) is the acceleration

In this exercise, equilibrium is reached because the rod travels at a constant velocity, meaning that \( a = 0 \). This implies all forces must balance so that the total net force is zero. Gravity pulling the rod down the incline is counteracted by the magnetic force from the current through the rod.

When we apply Newton's second law, the downward component of the gravitational force \( mg\sin\alpha \) matches with \( IlB \), the magnetic force. It's a perfect demonstration of how forces balance out, resulting in no acceleration. This balance allows us to derive the current necessary, using the equation: \[ I = \frac{mg\sin\alpha}{lB} \] Substituting the given values (\( m = 0.20 \text{ kg}, g = 9.8 \text{ m/s}^2, \sin 30.0^{\circ} = 0.5, l = 1.6 \text{ m}, B = 0.050 \text{ T} \)), we can calculate the current \( I \), showcasing how forces interplay in a real-world system.