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Derive the expression for torque acting on a dipole?
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Derive the expression for torque acting on a dipole?
Consider an electric dipole placed in a uniform electric field of strength E in such a way that its dipole moment vector p makes an angle q with the direction of vector E. The charges of dipole are - q and + q at separation 2l the dipole moment of electric dipole,
p = q.2l ...(1)
Force: The force on charge + q is, vector F1 = q vector E , along the direction of field vector E
The force on charge - q is, vector F2 = qE , opposite to the direction of field vector E
Obviously forces vector F1 and F2 are equal in magnitude but opposite in direction; hence net force on electric dipole in uniform electric field is
F = F1 - F2 = qE - qE = 0 (zero)
As net force on electric dipole is zero, so dipole does not undergo any translatory motion.
Torque: The forces vector F1 and vector F2 form a couple (or torque) which tends to rotate and align the dipole along the direction of electric field. This couple is called the torque and is denoted by τ.
∴ torque τ = magnitude of one force ' perpendicular distance between lines of action of forces
= qE (BN)
= qE (2lsinθ)
=(q 2l) Esinθ
= pEsinq [usinθ(1)] ....(2)
Clearly, the magnitude of torque depends on orientation (θ) of the electric dipole relative to electric field. Torque (τ) is a vector quantity whose direction is perpendicular to both vector p and vector E .
In vector form vector τ = vector p x vector E ...(3)
Thus, if an electric dipole is placed in an electric field in oblique orientation, it experiences no force but experiences a torque. The torque tends to align the dipole moment along the direction of electric field.
Maximum Torque: For maximum torque sinθ should be the maximum. As the maximum value of sin q =1 when θ = 90degree
∴ Maximum Torque, τmax = pE
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Derive the expression for torque acting on a dipole?
Expression for Torque Acting on a Dipole:
The torque acting on a dipole can be derived by considering the interaction between the electric field and the dipole moment. The torque is a measure of the rotational force experienced by the dipole when placed in an electric field.
1. Electric Field:
The electric field at any point in space is the force experienced by a unit positive charge placed at that point. It is denoted by E and has both magnitude and direction.
2. Dipole Moment:
A dipole is a pair of equal and opposite charges separated by a small distance. The dipole moment is defined as the product of the magnitude of one of the charges and the distance between them. It is denoted by p and has both magnitude and direction.
3. Torque:
Torque is the rotational analogue of force. It is the measure of the tendency of a force to cause an object to rotate about a specific axis. The torque acting on a dipole in an electric field is given by the equation:
τ = p * E * sin(θ)
where τ is the torque acting on the dipole, p is the magnitude of the dipole moment, E is the magnitude of the electric field, and θ is the angle between the dipole moment and the electric field vector.
4. Derivation:
To derive this expression, we consider the work done by the electric field on the dipole. The work done is given by the equation:
W = -p * E * cos(θ)
where W is the work done, p is the magnitude of the dipole moment, E is the magnitude of the electric field, and θ is the angle between the dipole moment and the electric field vector.
Since torque is defined as the rate of change of work, we differentiate the work equation with respect to the angle θ:
dW/dθ = p * E * sin(θ)
This expression represents the rate of change of work done by the electric field with respect to the angle θ. By definition, the rate of change of work is equal to the torque acting on the dipole:
τ = dW/dθ
Thus, the derived expression for torque acting on a dipole is:
τ = p * E * sin(θ)
Summary:
The torque acting on a dipole in an electric field is given by the equation τ = p * E * sin(θ), where p is the magnitude of the dipole moment, E is the magnitude of the electric field, and θ is the angle between the dipole moment and the electric field vector. This expression is derived by considering the work done by the electric field on the dipole and the rate of change of work, which is equal to the torque.
The torque acting on a dipole can be derived by considering the interaction between the electric field and the dipole moment. The torque is a measure of the rotational force experienced by the dipole when placed in an electric field.
1. Electric Field:
The electric field at any point in space is the force experienced by a unit positive charge placed at that point. It is denoted by E and has both magnitude and direction.
2. Dipole Moment:
A dipole is a pair of equal and opposite charges separated by a small distance. The dipole moment is defined as the product of the magnitude of one of the charges and the distance between them. It is denoted by p and has both magnitude and direction.
3. Torque:
Torque is the rotational analogue of force. It is the measure of the tendency of a force to cause an object to rotate about a specific axis. The torque acting on a dipole in an electric field is given by the equation:
τ = p * E * sin(θ)
where τ is the torque acting on the dipole, p is the magnitude of the dipole moment, E is the magnitude of the electric field, and θ is the angle between the dipole moment and the electric field vector.
4. Derivation:
To derive this expression, we consider the work done by the electric field on the dipole. The work done is given by the equation:
W = -p * E * cos(θ)
where W is the work done, p is the magnitude of the dipole moment, E is the magnitude of the electric field, and θ is the angle between the dipole moment and the electric field vector.
Since torque is defined as the rate of change of work, we differentiate the work equation with respect to the angle θ:
dW/dθ = p * E * sin(θ)
This expression represents the rate of change of work done by the electric field with respect to the angle θ. By definition, the rate of change of work is equal to the torque acting on the dipole:
τ = dW/dθ
Thus, the derived expression for torque acting on a dipole is:
τ = p * E * sin(θ)
Summary:
The torque acting on a dipole in an electric field is given by the equation τ = p * E * sin(θ), where p is the magnitude of the dipole moment, E is the magnitude of the electric field, and θ is the angle between the dipole moment and the electric field vector. This expression is derived by considering the work done by the electric field on the dipole and the rate of change of work, which is equal to the torque.
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Derive the expression for torque acting on a dipole?
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Derive the expression for torque acting on a dipole? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about Derive the expression for torque acting on a dipole? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Derive the expression for torque acting on a dipole?.
Derive the expression for torque acting on a dipole? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about Derive the expression for torque acting on a dipole? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Derive the expression for torque acting on a dipole?.
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