Q. Consider the equation: H=xpϵqErtsH = \frac{x^p \epsilon^q E^r}{t^s}H=tsxpϵqEr​ Where: H = magnetic field E = electric field ε = permittivity x = distance t = time Find the values of p, q, r and s respectively.

Consider the equation  H = (x^p ε^q E^r) / t^s

 

Where:  H = magnetic field,  E = electric field,  ε = permittivity,  x = distance,  t = time

 

Find the values of p, q, r and s.

 

Step 1: Dimensions of Each Quantity

Quantity	Dimensions
H  (Magnetic field)	[M T⁻² A⁻¹]
E  (Electric field)	[M L T⁻³ A⁻¹]
ε  (Permittivity)	[M⁻¹ L⁻³ T⁴ A²]
x  (Distance)	[L]
t  (Time)	[T]

 

Step 2: Dimensional Equation

Substituting dimensions into the equation:

 

[M T⁻² A⁻¹] = [L]ᵖ × [M⁻¹L⁻³T⁴A²]ᶠ × [MLT⁻³A⁻¹]ʳ × [T]⁻ˢ

 

Expanding: [M¹ T^-2 A^-1] = [M^-q+r L^p-3q+r T^4q-3r-s A^2q-r]

 

Step 3: Compare Powers on Both Sides

Element	Equation	Result
M	-q + r = 1	Equation (1)
A	2q - r = -1	Equation (2)
L	p - 3q + r = 0	Equation (3)
T	4q - 3r - s = -2	Equation (4)

 

Step 4: Solve the Equations

From equations (1) and (2) — adding both:

-q + r = 1

2q - r = -1

∴  q = 0

Substituting q = 0 in (1):  r = 1

 

From equation (3):

p - 3(0) + 1 = 0   ⇒   p = -1

 

From equation (4):

4(0) - 3(1) - s = -2   ⇒   -3 - s = -2   ⇒   s = -1

 

 

Final Answer

p = −1,    q = 0,    r = 1,    s = −1

 

Verification

Substituting back:  H = (x⁻¹ ε⁰ E¹) / t⁻¹  =  Et/x