28.8
Magnetische kracht op stroomvoerende draden: voorbeeld
In een magnetisch veld ondervinden bewegende ladingen een kracht. Als een draad deze bewegende ladingen bevat, d.w.z. als de draad stroom voert, dan w…
Consider a conductor in the XY plane carrying current to the left. The conductor consists of two straight segments with a length of L parallel to the x-axis and a semicircle with a radius of R.
If the conductor is placed in a uniform magnetic field perpendicular to the plane of the wire, pointing outward, determine the total magnetic force on the conductor.
Firstly, determine the magnetic force for the individual straight sections carrying current in the x direction.
Secondly, divide the curved segment into infinitesimal straight sections having a length of dl at an angle of theta.
The infinitesimal length and the magnetic field are perpendicular to each other, and the direction of their cross product is radially outward. So, the magnitude and components of the force on the sections can be determined.
By integrating the expression with respect to theta, the force on the whole curved conductor can be determined.
Finally, by adding the forces on all three sections, the total force on the conductor in the positive y direction can be obtained.
View the full transcript and gain access to JoVE Core videos
Q1: How does a magnetic force act on a current-carrying conductor?
A magnetic force acts on a current-carrying conductor when it is placed in a magnetic field. Moving charges in the wire experience a force perpendicular to both the current direction and the magnetic field. The magnitude of this force depends on the current strength, conductor length, magnetic field strength, and the angle between them. Using the right-hand rule helps determine the force direction.
Q2: What is the relationship between current, magnetic field, and force direction?
The magnetic force on a current-carrying conductor is determined by the cross product of current direction and magnetic field direction. The right-hand rule provides a quick method: point your fingers along the current direction, curl them toward the magnetic field direction, and your thumb points in the force direction. This perpendicular relationship ensures the force acts at 90 degrees to both vectors.
Q3: How do you calculate the total force on a conductor with multiple segments?
To find the total force on a conductor with straight and curved segments, calculate the force on each section separately, then add them vectorially. For straight segments carrying current in a uniform field, use F = IL × B. For curved segments, divide them into infinitesimal elements and integrate the force contributions across the entire curve to obtain the total force.
Q4: Why does a magnetic field create a force that can balance gravitational force?
A sufficiently strong magnetic field creates an upward force on a current-carrying wire that can equal and oppose the downward gravitational force. When these forces balance, the tension in supporting leads becomes zero and the wire achieves static equilibrium. The magnetic force magnitude depends on current strength and field strength, allowing precise force control through current adjustment.
Q5: What role does the angle between current and magnetic field play in force magnitude?
The magnetic force magnitude depends on the sine of the angle between current direction and magnetic field direction. When perpendicular, the angle is 90 degrees and force is maximum. For curved conductors, different segments make different angles with the field, so their force components vary. Integration accounts for these varying angles across the curved path.
Q6: How do you determine current needed to counteract wire weight in a magnetic field?
To find the required current, set up a free-body diagram balancing gravitational and magnetic forces. Use the right-hand rule to confirm the magnetic force points upward. Then equate the weight force to the magnetic force expression: mg = BIL. Solving for current gives I = mg/(BL), where m is mass, g is gravitational acceleration, B is field strength, and L is wire length.
Q7: How does integration help calculate force on curved current-carrying segments?
For curved segments, divide the path into infinitesimal elements of length dl at varying angles to the magnetic field. Each element experiences a small force dF = I(dl × B). Integrating this expression over the entire curve from zero to the total angle yields the total force on the curved section. This method handles the changing angle and direction of force components along the curve.