Question

Intensity the correct order for electric field due to Electric dipole at a point P along the dipole axis at distance \(Z\) from dipole's midpoint when
• \(Z\) is halved
• \(Z\) is doubled
• \(Z\) remains same
• \(Z\) is made three times Choose the correct answer from the options given below:

• A. D $<$ B $<$ C $<$ A
• B. B $<$ A $<$ D $<$ C
• C. A $<$ C $<$ B $<$ D
• D. C $<$ D $<$ A $<$ B

Hint:

Dipole electric field varies as: \[ E\propto \frac{1}{r^3} \] Doubling distance reduces field by: \[ \frac18 \] and halving distance increases field by: \[ 8 \] times.

Answer 1

The Correct Option is A

Concept: Electric field due to a dipole along its axial line varies inversely as the cube of distance. The relation is: :contentReference[oaicite:0]{index=0} Thus: \[ \boxed{ \text{Smaller distance} \Rightarrow \text{Stronger electric field} } \] and: \[ \boxed{ \text{Larger distance} \Rightarrow \text{Weaker electric field} } \]

Step 1: Write the electric field expression. For a dipole along axial line: :contentReference[oaicite:1]{index=1} Thus: \[ E \propto \frac{1}{r^3} \]

Step 2: Find field when distance is halved. If: \[ r=\frac{Z}{2} \] then: \[ E\propto \frac{1}{(Z/2)^3} \] \[ E\propto \frac{8}{Z^3} \] Thus electric field becomes: \[ \boxed{ 8 \text{ times} } \] This is the largest value. Hence: \[ A=\text{maximum} \]

Step 3: Find field when distance is doubled. If: \[ r=2Z \] then: \[ E\propto \frac{1}{(2Z)^3} \] \[ E\propto \frac{1}{8Z^3} \] Thus field becomes: \[ \boxed{ \frac18 } \] of original value.

Step 4: Find field when distance remains same. If: \[ r=Z \] then: \[ E\propto \frac{1}{Z^3} \] This is the reference value.

Step 5: Find field when distance becomes three times. If: \[ r=3Z \] then: \[ E\propto \frac{1}{(3Z)^3} \] \[ E\propto \frac{1}{27Z^3} \] This is the smallest value. Hence: \[ D=\text{minimum} \]

Step 6: Arrange in increasing order. Smallest to largest: \[ D < B < C < A \]

Step 7: Choose the correct answer. Thus the correct option is: \[ \boxed{ D < B < C < A } \] Hence the correct option is: \[ \boxed{(1)} \] Final Conclusion: Since: \[ E\propto \frac{1}{r^3} \] the correct increasing order is: \[ \boxed{ D < B < C < A } \] Hence, the correct answer is: \[ \boxed{(1)} \]