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Electrostatics question

2022 · 28 Jul · Shift 1 · Q67
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  5. /2022 · 28 Jul · Shift 1 · Q67

Electrostatics question

2022 · 28 Jul · Shift 1 · Q67

JEE MainPhysicsElectrostaticsNumerical+4 / −1
Two electric dipoles of dipole moments 1.2×10−30 Cm1.2 \times 10^{-30} \,\mathrm{Cm}1.2×10−30Cm and 2.4×10−30 Cm2.4 \times 10^{-30} \,\mathrm{Cm}2.4×10−30Cm are placed in two different uniform electric fields of strengths 5×104 NC−15 \times 10^{4} \,\mathrm{NC}^{-1}5×104NC−1 and 15×104 NC−115 \times 10^{4} \,\mathrm{NC}^{-1}15×104NC−1 respectively. The ratio of maximum torque experienced by the electric dipoles will be 1x\frac{1}{x}x1​. The value of xxx is ‾\underline{\hspace{2cm}}​.
Numerical answer
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Correct answer: 6

  1. The torque on an electric dipole in a uniform electric field is
τ=pEsin⁡θ\tau = pE\sin\thetaτ=pEsinθ

where ppp is dipole moment, EEE is electric field, and θ\thetaθ is the angle between them.

  1. The maximum torque occurs when
sin⁡θ=1⇒θ=90∘\sin\theta = 1 \quad \Rightarrow \quad \theta = 90^\circsinθ=1⇒θ=90∘

So,

τmax⁡=pE\tau_{\max} = pEτmax​=pE
  1. For the first dipole:
p1=1.2×10−30 Cm,E1=5×104 N/Cp_1 = 1.2 \times 10^{-30}\,\text{Cm}, \qquad E_1 = 5 \times 10^4\,\text{N/C}p1​=1.2×10−30Cm,E1​=5×104N/C

Thus,

τ1=p1E1=(1.2×10−30)(5×104)\tau_1 = p_1E_1 = (1.2 \times 10^{-30})(5 \times 10^4)τ1​=p1​E1​=(1.2×10−30)(5×104) τ1=6×10−26 N m\tau_1 = 6 \times 10^{-26}\,\text{N m}τ1​=6×10−26N m
  1. For the second dipole:
p2=2.4×10−30 Cm,E2=15×104 N/Cp_2 = 2.4 \times 10^{-30}\,\text{Cm}, \qquad E_2 = 15 \times 10^4\,\text{N/C}p2​=2.4×10−30Cm,E2​=15×104N/C

Thus,

τ2=p2E2=(2.4×10−30)(15×104)\tau_2 = p_2E_2 = (2.4 \times 10^{-30})(15 \times 10^4)τ2​=p2​E2​=(2.4×10−30)(15×104) τ2=36×10−26 N m\tau_2 = 36 \times 10^{-26}\,\text{N m}τ2​=36×10−26N m
  1. Ratio of maximum torques:
τ1:τ2=6×10−26:36×10−26=1:6\tau_1 : \tau_2 = 6 \times 10^{-26} : 36 \times 10^{-26} = 1:6τ1​:τ2​=6×10−26:36×10−26=1:6

So,

τ1τ2=16\frac{\tau_1}{\tau_2} = \frac{1}{6}τ2​τ1​​=61​

Given that the ratio is 1x\dfrac{1}{x}x1​,

1x=16\frac{1}{x} = \frac{1}{6}x1​=61​

Hence,

x=6x = 6x=6
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