Force on a Current Carrying Conductor in Magnetic Field

Force on a Current-Carrying Conductor is a fundamental concept in the field of electromagnetism. It was discovered by Hans Christian Oersted in the year 1820 that when current flows through a conductor, it produces a magnetic field around it. When this conductor is placed in a uniform magnetic field, the magnetic field produced by it interacts with the external magnetic field to produce a force on the conductor which we will learn about it in detail in this article further. This article covers the topic of  Force on a Current-Carrying Conductor and its mathematical formula in detail.  Also the rule for finding the direction of the motion due to this force.

Magnetic Force

Magnetic Force is the force exerted by a magnet or a moving charge on another magnet or moving charge in their region of the magnetic field. If we place a moving charge in the magnetic field region of another moving charge then it experiences both electric and magnetic fields. This is because a moving charge is the source of both an electric field and a magnetic field. Hence, the total force exerted by a charge in the magnetic field created by a moving charge is given as the sum of electric force and magnetic force i.e. F_magnetic + F_electric. The force was given by Lorentz which is discussed in the below topic. 

Lorentz Force

Force acting on a charge moving in the magnetic field when there is no electric field acting on the conductor, is called Lorentz force, and the mathematical expression for the Lorentz Force is given as follows:

OR

F = qvBsinθ 

where ,

• q is the charge, 
• v is the velocity of the charge moving in a magnetic field, 
• B is the magnetic field, and 
• θ is the angle between the velocity of charge and magnetic field.

What is the Force on Current Carrying Conductor in Magnetic Field?

When a current-carrying conductor is placed in a uniform magnetic field, then the moving charges inside the conductor experience the force (Due to Lorentz force), this is the required force on a current-carrying conductor. For calculating this force let us consider the various parameters as follows:

• L is the length of the conductor, 
• I is the current flowing through it, 
• q is the charges flowing through the conductor at the time ‘t‘, 
• v is the velocity of the charge q, 
• B is a Uniform magnetic field in which current carrying conductor is placed

Then, according to Lorentz Force

F = qvBsinθ    [ Now, q = I × t, and v = L / t ]

F = (I × t) × (L/t) × Bsinθ

Thus, F = BILsinθ

Where, 

L is the length of the conductor, 

I is current flowing through it, 

B is a Uniform magnetic field in which current carrying conductor is placed, and 

θ is the angle between the velcity of charge and magnetic field.

Vector Form

The vector form of the formula for finding force on the current-carrying conductor is given by, 

Special Cases of Force on Current Carrying Conductor

There are a few special cases of force on the current-carrying conductors depending on the position of the conductor in the magnetic field. These cases are explained as follows:

Case I: When a conductor is placed parallel to the Magnetic Field

When sin θ = 0 (minimum) i.e., θ = 0° or 180°, then force on the current element in a magnetic field is zero(minimum).

F_min = 0

A current element in a magnetic field does not experience any force if the current in it is collinear with the field, thus it is the least magnitude of the force experienced by the conductor in the given magnetic field.

Case II: When a conductor is placed perpendicular to the Magnetic Field

When sin θ = 1 (maximum) i.e., θ = 90°, then force on the current element in a magnetic field is maximum (=ILB).

F_max = ILB

The direction of force is always perpendicular to the plane containing  and   and this is the maximum possible force experienced by the conductor in the given magnetic field.

Direction of Force on a Current-Carrying Conductor

The direction of the force on a current-carrying conductor can be determined using the left and right-hand rules or thumb rules, which are explained as follows:

Fleming’s Left-Hand Rule

Fleming’s Left-Hand Rule states that if we stretch the forefinger, middle finger, and thumb perpendicularly to each other then the forefinger shows the direction of the magnetic field, the middle finger shows the direction of current in the conductor and the thumb shows the direction of the motion of the conductor. This law is generally used to find the direction of the magnetic field if the motion of the conductor is given. i

Fleming’s Right-Hand Palm Rule

Another important rule which is used in electrostatics is Right-Hand Palm Rule. It says if we, stretch the fingers and thumb of the right hand at right angles to each other. Then, if the fingers point in the direction of the field  and the thumb in the direction of the current I, the normal to palm will point in the direction of the force. This law is used to find the induced EMF or current in the conductor when placed in a varying magnetic field.

Solved Examples on Force on Current Carrying Conductor in Magnetic Field

Example 1: A current of 1A flows in a wire of length 0.1cm in a magnetic field of 0.5T. Calculate the force acting on the wire when the wire makes an angle of (i) 90° (ii) 0° with respect to the magnetic field. 

Solution: 

(i) For an angle of 90°

F = ILBsinθ
⇒ F = 1×0.1×0.5×sin90°
⇒ F = 0.05 N  

(ii) For angle 0°

F = ILBsinθ 
⇒ F = 1×0.1×0.5×sin0°
⇒ F = 0N

Example 2: A current-carrying conductor of length 0.5cm with current 2A is placed at an angle of 30° in the magnetic field of 0.3T. Calculate the force acting on it.

Solution: 

Given: L = 0.5 cm, I = 2A, B = 0.3T and θ = 30°

F = ILBsinθ
⇒ F = 2×0.5×0.3×sin30°
⇒ F = 0.3/2

⇒ F = 0.15N

Example 3: Find the length of the current-carrying conductor with 3A current which is placed at 90° in the magnetic field of 0.5T with 0.3 N force acting on it.

Solution: 

Given: I = 3A, B = 0.5T, F =0.3N and θ = 90°

F = ILBsinθ

⇒ 0.3 = 3×L×0.5×sin90°

⇒ L = 0.3/1.5

⇒ L = 0.2cm

Example 4: At what angle the current-carrying conductor of length 0.6cm with a current 2A is placed in the magnetic field of 0.2T with 0.24N force acting on it?

Solution: 

Given:L = 0.6cm, I = 2A, B = 0.2T and F=0.24N

F = ILBsinθ

⇒ 0.24 = 2×0.6×0.2×sinθ

⇒ 1 = sinθ

⇒ θ = 90°   

FAQs on Force on Current Carrying Conductor

Q1: What is Fleming’s Left Hand Rule?

Answer: 

Fleming’s Left Hand Rule states that,

“Stretch the forefinger, central finger, and thumb of the left hand mutually perpendicular.Then if the forefinger points in the direction of the field (), the central finger in the direction of current I, the thumb will point in the direction of the force.”

Q2: For Maximum Force acting on the Current-Carrying Conductor what is the Equation of Force?

Answer: 

The force acting on the current-carrying conductor is maximum when θ = 90° and the equation of force is

F = ILB.

Q3: What is the Vector Form for the Force Acting on the Current-Carrying Conductor?

Answer:

Vector form for the force acting on the current-carrying conductor is given by:

Q4: What is the Condition for which the Force on the Current-Carrying Conductor is Zero?

Answer: 

When the angle between the length of the current-carrying conductor and the magnetic field is 0° i.e., both current-carrying conductors are collinear then force F =0.

Q5: What is Lorentz’s Force?

Answer:

Force acting on a charge moving in a magnetic field when influence of electric filed on the conductor is zero, is called Lorentz force. It is given by:

OR

F = qvBsinθ