Question

The displacement current term was added to Ampere's law by Maxwell to ensure
• Conservation of energy
• Consistency with the continuity equation
• The speed of light is constant
• Magnetic fields are circular

• A. Conservation of energy
• B. Consistency with the continuity equation
• C. The speed of light is constant
• D. Magnetic fields are circular

Hint:

Displacement current was introduced by Maxwell to make Ampere's law compatible with: \[ \boxed{ \text{Continuity equation} } \] and charge conservation.

Answer 1

The Correct Option is B

Concept: Ampere's circuital law originally related magnetic field to electric current. However, Maxwell discovered that the original law fails for time-varying electric fields. To correct this inconsistency, he introduced: \[ \boxed{ \text{Displacement current} } \]

Step 1: Write original Ampere's law. Original Ampere's law is: :contentReference[oaicite:2]{index=2} where:
• \(\vec B\) = magnetic field
• \(\vec J\) = conduction current density

Step 2: Understand the problem. Taking divergence on both sides: \[ \nabla\cdot(\nabla\times\vec B)=0 \] Thus: \[ \nabla\cdot\vec J=0 \] But continuity equation states: :contentReference[oaicite:3]{index=3} For changing charge density: \[ \frac{\partial\rho}{\partial t}\neq0 \] Hence original Ampere's law becomes inconsistent.

Step 3: Introduce displacement current. Maxwell modified Ampere's law to: :contentReference[oaicite:4]{index=4} The second term: \[ \epsilon_0\frac{\partial\vec E}{\partial t} \] is called displacement current density.

Step 4: Understand why Maxwell added this term. The displacement current term ensures:
• Charge conservation
• Compatibility with continuity equation Hence: \[ \boxed{ \text{Consistency with continuity equation} } \]

Step 5: Analyze the options.
• Conservation of energy \(\rightarrow\) not the primary reason
• Consistency with continuity equation \(\rightarrow\) correct
• Speed of light constant \(\rightarrow\) consequence later
• Magnetic fields circular \(\rightarrow\) unrelated

Step 6: Choose the correct answer. Thus the correct answer is: \[ \boxed{ \text{Consistency with the continuity equation} } \] Hence the correct option is: \[ \boxed{(2)} \] Final Conclusion: Maxwell introduced displacement current to maintain: \[ \boxed{ \text{Consistency with charge conservation and continuity equation} } \] Hence, the correct answer is: \[ \boxed{(2)} \]