एक इलेक्ट्रॉन एवं एक प्रोटॉन के लिए स्थिरवैद्युत बल एवं गुरुत्वाकर्षण बल के परिमाण का अनुपात क्या है?

Physics
एक इलेक्ट्रॉन एवं एक प्रोटॉन के लिए स्थिरवैद्युत बल एवं गुरुत्वाकर्षण बल के परिमाण का अनुपात क्या है?

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Answer

(b)

Step 1: The magnitude of the electrostatic force between an electron and a proton is Fe=kqe2r2F_e = \frac{k q_e^2}{r^2}, and the gravitational force is Fg=Gmempr2F_g = \frac{G m_e m_p}{r^2}.

The ratio is

FeFg=kqe2Gmemp\frac{F_e}{F_g} = \frac{k q_e^2}{G m_e m_p}

where k=9×109 N⋅m2/C2k = 9 \times 10^9 \, \mathrm{N \cdot m^2 / C^2}, qe=1.6×10−19 Cq_e = 1.6 \times 10^{-19} \, \mathrm{C}, G=6.7×10−11 N⋅m2/kg2G = 6.7 \times 10^{-11} \, \mathrm{N \cdot m^2 / kg^2}, me=9.1×10−31 kgm_e = 9.1 \times 10^{-31} \, \mathrm{kg}, mp=1.67×10−27 kgm_p = 1.67 \times 10^{-27} \, \mathrm{kg}.

Step 2: Compute qe2q_e^2:

qe2=(1.6×10−19)2=2.56×10−38 C2.q_e^2 = (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \, C^2.

Numerator: kqe2=9×109×2.56×10−38=23.04×10−29=2.304×10−28 N⋅m2k q_e^2 = 9 \times 10^9 \times 2.56 \times 10^{-38} = 23.04 \times 10^{-29} = 2.304 \times 10^{-28} \, \mathrm{N \cdot m^2}.

Step 3: Compute mempm_e m_p:

memp=9.1×10−31×1.67×10−27=15.197×10−58=1.52×10−57 kg2.m_e m_p = 9.1 \times 10^{-31} \times 1.67 \times 10^{-27} = 15.197 \times 10^{-58} = 1.52 \times 10^{-57} \, kg^2.

Denominator: Gmemp=6.7×10−11×1.52×10−57=10.184×10−68=1.02×10−67 N⋅m2G m_e m_p = 6.7 \times 10^{-11} \times 1.52 \times 10^{-57} = 10.184 \times 10^{-68} = 1.02 \times 10^{-67} \, \mathrm{N \cdot m^2}.

Step 4: Compute the ratio:

FeFg=2.304×10−281.02×10−67=2.3041.02×10−28−(−67)=2.26×1039≈2.3×1039.\frac{F_e}{F_g} = \frac{2.304 \times 10^{-28}}{1.02 \times 10^{-67}} = \frac{2.304}{1.02} \times 10^{-28 - (-67)} = 2.26 \times 10^{39} \approx 2.3 \times 10^{39}.

(b) 2.3×10392.3 \times 10^{39}

(b)

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Quick Answer

The magnitude of the electrostatic force between an electron and a proton is F_e = (k q_e^2)/(r^2), and the gravitational force is F_g = (G m_e m_p)/(r^2).

एक इलेक्ट्रॉन एवं एक प्रोटॉन के लिए स्थिरवैद्युत बल एवं गुरुत्वाकर्षण बल के परिमाण का अनुपात क्या है?
Physics

This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.

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Step 1: The magnitude of the electrostatic force between an electron and a proton is F_e = (k q_e^2)/(r^2), and the gravitational force is F_g = (G m_e m_p)/(r^2). The ratio is (F_e)/(F_g) = (k q_e^2)/(G m_e m_p) where k = 9 × 10^9 \, N · m^2 / C^2, q_e = 1.6 × 10^-19 \, C, G = 6.7 × 10^-11 \, N · m^2 / kg^2, m_e = 9.1 × 10^-31 \, kg, m_p = 1.67 × 10^-27 \, kg. Step 2: Compute q_e^2: q_e^2 = (1.6 × 10^-19)^2 = 2.56 × 10^-38 \, C^2. Numerator: k q_e^2 = 9 × 10^9 × 2.56 × 10^-38 = 23.04 × 10^-29 = 2.304 × 10^-28 \, N · m^2. Step 3: Compute m_e m_p: m_e m_p = 9.1 × 10^-31 × 1.67 × 10^-27 = 15.197 × 10^-58 = 1.52 × 10^-57 \, kg^2. Denominator: G m_e m_p = 6.7 × 10^-11 × 1.52 × 10^-57 = 10.184 × 10^-68 = 1.02 × 10^-67 \, N · m^2. Step 4: Compute the ratio: (F_e)/(F_g) = 2.304 × 10^-281.02 × 10^-67 = (2.304)/(1.02) × 10^-28 - (-67) = 2.26 × 10^39 ≈ 2.3 × 10^39. (b) 2.3 × 10^39 (b)