Let's break down the scenario you've presented regarding the parallel plate capacitor immersed in a conducting liquid and subjected to a magnetic field. This situation involves several physical principles, including electromagnetism, fluid dynamics, and circuit theory. Each option you've listed can be analyzed based on these principles.

Understanding the System

We have a parallel plate capacitor with plate area A and separation d, placed in a conducting liquid that flows with a constant velocity V. The liquid has a conductivity s, and there's a magnetic field B that is perpendicular to the flow of the liquid. The capacitor is connected to an external resistance R. Let's evaluate each statement one by one.

Evaluating Each Statement

• a. The current in the circuit is constant

In this scenario, the flow of the conducting liquid creates a motion-induced electromotive force (emf) due to the interaction with the magnetic field. According to Faraday's law of electromagnetic induction, a changing magnetic field or a conductor moving through a magnetic field induces an emf. However, since the liquid flows at a constant velocity and the magnetic field is constant, the induced emf will also be constant. This means that the current in the circuit can be considered constant, assuming the resistance R does not change. Therefore, this statement is true.

• b. Magnetic field causes work which causes heat liberation

The interaction between the magnetic field and the moving conductive liquid does indeed involve work being done. The Lorentz force acts on the charges in the liquid, causing them to move and generate current. This movement of charges through the resistance R results in power dissipation in the form of heat, as described by Joule's law (P = I²R). Thus, this statement is also true.

• c. The power dissipated in external resistance is (V² B² d² R)/(d/A + R)

To analyze this statement, we need to consider the power dissipated in the resistor. The induced emf (E) due to the motion of the liquid can be expressed as E = B * V * d, where B is the magnetic field strength, V is the velocity of the liquid, and d is the separation between the plates. The current (I) through the resistor can be calculated as I = E / R. The power (P) dissipated in the resistor is given by P = I²R. Substituting the expression for I, we can derive the power formula. However, the provided formula does not seem to match the standard derivation, indicating that this statement is likely false.

• d. External forces are to be applied to maintain constant flow of liquid with uniform speed

In a real-world scenario, to maintain a constant flow of a conducting liquid, external forces may indeed be required to counteract any frictional forces or drag that arise due to the viscosity of the liquid and the interaction with the capacitor plates. If the liquid is flowing steadily, it implies that the net force acting on it is zero, which means that any driving force must balance the resistive forces. Therefore, this statement is true.

Summary of Findings

Based on our analysis, we can conclude the following:

• Statement a is true: The current in the circuit is constant.
• Statement b is true: The magnetic field causes work, leading to heat liberation.
• Statement c is likely false: The power formula provided does not align with standard derivations.
• Statement d is true: External forces are necessary to maintain the constant flow of the liquid.

Understanding these principles not only helps in solving this particular question but also builds a foundation for exploring more complex systems involving electromagnetism and fluid dynamics. If you have any further questions or need clarification on any point, feel free to ask!