The Question is from neet-physics magnetic-effects-current-and-magnetism , and it was created by the author P.NARASIMHA

Question

If ${{m{	heta }}_1}{m{ \;and\; }}{{m{	heta }}_2}$ be the apparent angles of dip observed in two vertical planes at right angles to each other, then the true angle of dip ${m{	heta }}$ is given by

P.NARASIMHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: [B]
## CONCEPT FRAMEWORK
	**Primary Concept**: Relationship between true and apparent angles of dip
	**Related Topics**: Magnetic dip, Trigonometric identities, Vector components
	**Prerequisites**: Understanding of magnetic dip, basic trigonometry
## SOLUTION PATHWAY
**Given Information**:
	*  Apparent dip angles: θ₁ and θ₂
	*  Planes are perpendicular to each other
	*  True dip angle: θ
**Method & Steps**:
	1.  Represent the true magnetic field vector as a combination of horizontal (H) and vertical (V) components.
		Reasoning: The magnetic field vector can be resolved into horizontal and vertical components.
		Formula/Rule: B = √(H² + V²)
	2.  Relate the apparent angles to the vertical component and the horizontal component in each plane.
		Working: V = H tan(θ), V = H₁ tan(θ₁), V = H₂ tan(θ₂). Also, H² = H₁² + H₂²
		Key Point: The vertical component is the same in all planes. The horizontal component gets resolved.
	3. Express the true angle in terms of the apparent angles.
		Working:  cot²(θ) = (H²/V²) = (H₁² + H₂²)/V² = H₁²/V² + H₂²/V² = cot²(θ₁) + cot²(θ₂)
		Result:  cot²(θ) = cot²(θ₁) + cot²(θ₂)
		Verification: The derived result matches option B.
 ## OPTION ANALYSIS
[A] :x: This option uses tan² and a subtraction which is not the relation between true and apparent dips.
[B] ✓: This is the correct relationship between the true dip angle and the apparent dip angles in two perpendicular planes.
[C] :x: This option uses tan² and an addition instead of cot² and addition.
[D] :x: This option uses cot² and a subtraction instead of cot² and addition.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Confusing tan and cot and their relationships. [How to avoid]: Remember that cot is the reciprocal of tan and the relationship is based on cot.
		2.[Error 2]: Incorrectly resolving the horizontal component. [How to avoid]: Visualize the horizontal component of the magnetic field as a vector and understand how it gets resolved in two perpendicular planes.
	**Quick Tips**:
		1.[Helpful shortcut]: Remember the relation is between cot² of the true angle and the sum of cot² of apparent angles.
		2.[Memory aid]: Think of the horizontal component as a vector sum of its components in the perpendicular planes, and the vertical component remains the same.

P.NARASIMHA · Answer

${	an ^2}{m{	heta }} = {	an ^2}{{m{	heta }}_1} - {	an ^2}{{m{	heta }}_2}$

P.NARASIMHA · Answer

The correct Option is "${\cot ^2}{m{	heta }} = {\cot ^2}{{m{	heta }}_1} + {\cot ^2}{{m{	heta }}_2}$"

P.NARASIMHA · Answer

${	an ^2}{m{	heta }} = {	an ^2}{{m{	heta }}_1} + {	an ^2}{{m{	heta }}_2}$

P.NARASIMHA · Answer

${\cot ^2}{m{	heta }} = {\cot ^2}{{m{	heta }}_1} - {\cot ^2}{{m{	heta }}_2}$

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