Continuous Charge Distribution

Continuous Charge Distribution

An infinite line charge has \( E = 7.2 \times 10^5 \, \text{N/C} \) at 5 cm. What is \( \lambda \)?

**Continuous distribution** uses linear density λ = dq/dl (C/m), surface σ = dq/dA, volume ρ = dq/dV. Field of infinite line with uniform λ is E = 2kλ/r = λ/(2π ε₀ r) radially outward, derived via cylindrical Gaussian surface, showing 1/r dependence. E = (2 k λ/r) . 7.2 × 10⁵ = (2 × 9 × 10⁹ × λ/0.05) . λ = (7.2 × 10⁵ × 0.05/18 × 10⁹) = 2 × 10⁻⁶ C/m . Substituting values gives 2.0 × 10⁻⁶ C/m, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

Ref: NCERT > Physics Book > Electric Charges and Fields > Continuous Charge Distribution

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Continuous Charge Distribution

An infinite line charge has \( E = 5.4 \times 10^5 \, \text{N/C} \) at 6 cm. What is \( \lambda \)?

**Continuous distribution** uses linear density λ = dq/dl (C/m), surface σ = dq/dA, volume ρ = dq/dV. Field of infinite line with uniform λ is E = 2kλ/r = λ/(2π ε₀ r) radially outward, derived via cylindrical Gaussian surface, showing 1/r dependence. E = (2 k λ/r) . 5.4 × 10⁵ = (2 × 9 × 10⁹ × λ/0.06) . λ = (5.4 × 10⁵ × 0.06/18 × 10⁹) = 1.8 × 10⁻⁶ C/m . Substituting values gives 1.8 × 10⁻⁶ C/m, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

Ref: NCERT > Physics Book > Electric Charges and Fields > Continuous Charge Distribution

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Continuous Charge Distribution

Why does the electric field remain constant inside a region where field lines are uniformly spaced and parallel?

**Continuous distribution** uses linear density λ = dq/dl (C/m), surface σ = dq/dA, volume ρ = dq/dV. Field of infinite line with uniform λ is E = 2kλ/r = λ/(2π ε₀ r) radially outward, derived via cylindrical Gaussian surface, showing 1/r dependence. Uniformly spaced, parallel field lines indicate a uniform field, where the field strength and direction do not vary. This occurs in regions like between parallel plates, where the field is constant due to consistent charge distribution. Substituting values gives Uniformity, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

Ref: NCERT > Physics Book > Electric Charges and Fields > Continuous Charge Distribution

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Continuous Charge Distribution

An infinite line charge has \( \lambda = 1.5 \times 10^{-6} \, \text{C/m} \). What is the electric field at 50 cm?

**Continuous distribution** uses linear density λ = dq/dl (C/m), surface σ = dq/dA, volume ρ = dq/dV. Field of infinite line with uniform λ is E = 2kλ/r = λ/(2π ε₀ r) radially outward, derived via cylindrical Gaussian surface, showing 1/r dependence. E = (2 k λ/r) , k = 9 × 10⁹ N·m²/C² . E = (2 × 9 × 10⁹ × 1.5 × 10⁻⁶/0.5) = 5.4 × 10⁴ N/C . Substituting values gives 5.4 × 10⁴ N/C, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

Ref: NCERT > Physics Book > Electric Charges and Fields > Continuous Charge Distribution

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Continuous Charge Distribution

A plane sheet has a surface charge density \( \sigma = 1.77 \times 10^{-10} \, \text{C/m}^2 \). What is the electric fie

**Line charge concept** extends point charge to infinite wire where symmetry dictates radial field proportional to λ and inversely proportional to distance r. λ = q/L for uniform case, field direction depends on sign of λ, outward for positive. E = (sigma/2 ε₀) . E = (1.77 × 10⁻¹⁰/2 × 8.854 × 10⁻¹²) = 10 N/C . Substituting values gives 10 N/C, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

Ref: NCERT > Physics Book > Electric Charges and Fields > Continuous Charge Distribution

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Continuous Charge Distribution

A conducting sphere of radius 30 cm has a surface charge density of \( 50 \, \mu\text{C/m}^2 \). What is the total charg

**Line charge concept** extends point charge to infinite wire where symmetry dictates radial field proportional to λ and inversely proportional to distance r. λ = q/L for uniform case, field direction depends on sign of λ, outward for positive. Surface area: A = 4 π r² = 4 π (0.3)² = 0.36 π m² . Charge: q = sigma A = 50 × 10⁻⁶ × 0.36 π = 5.654 × 10⁻⁵ C . Substituting values gives 5.65 × 10⁻⁵ C, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

Ref: NCERT > Physics Book > Electric Charges and Fields > Continuous Charge Distribution

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Continuous Charge Distribution

An infinite line charge has \( E = 6.0 \times 10^5 \, \text{N/C} \) at 3 cm. What is \( \lambda \)?

**Continuous distribution** uses linear density λ = dq/dl (C/m), surface σ = dq/dA, volume ρ = dq/dV. Field of infinite line with uniform λ is E = 2kλ/r = λ/(2π ε₀ r) radially outward, derived via cylindrical Gaussian surface, showing 1/r dependence. E = (2 k λ/r) . 6.0 × 10⁵ = (2 × 9 × 10⁹ × λ/0.03) . λ = (6.0 × 10⁵ × 0.03/18 × 10⁹) = 1 × 10⁻⁶ C/m . Substituting values gives 1.0 × 10⁻⁶ C/m, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

Ref: NCERT > Physics Book > Electric Charges and Fields > Continuous Charge Distribution

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Continuous Charge Distribution

An infinite line charge has \( E = 3.6 \times 10^5 \, \text{N/C} \) at 10 cm. What is \( \lambda \)?

**Line charge concept** extends point charge to infinite wire where symmetry dictates radial field proportional to λ and inversely proportional to distance r. λ = q/L for uniform case, field direction depends on sign of λ, outward for positive. E = (2 k λ/r) . 3.6 × 10⁵ = (2 × 9 × 10⁹ × λ/0.1) . λ = (3.6 × 10⁵ × 0.1/18 × 10⁹) = 2 × 10⁻⁶ C/m . Substituting values gives 2.0 × 10⁻⁶ C/m, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

Ref: NCERT > Physics Book > Electric Charges and Fields > Continuous Charge Distribution

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Continuous Charge Distribution

What causes the electric field to be discontinuous across a charged surface, such as a thin sheet?

**Line charge concept** extends point charge to infinite wire where symmetry dictates radial field proportional to λ and inversely proportional to distance r. λ = q/L for uniform case, field direction depends on sign of λ, outward for positive. The surface charge density on the sheet generates a field that changes abruptly. Gauss’s law shows the field on either side is sigma/2ε₀ , with opposite directions, resulting in a discontinuity equal to sigma/ε₀ across the surface. Substituting values gives Surface charge density, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

Ref: NCERT > Physics Book > Electric Charges and Fields > Continuous Charge Distribution

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Continuous Charge Distribution

An infinite line charge has \( \lambda = 5 \times 10^{-7} \, \text{C/m} \). What is the electric field at 10 cm?

**Charge density formulation** allows integration over extended bodies, but highly symmetric cases yield simple expressions. Infinite line gives E ∝ λ/r, unlike point charge 1/r², reflecting different geometry of source. E = (2 k λ/r) . E = (2 × 9 × 10⁹ × 5 × 10⁻⁷/0.1) = 9 × 10⁴ N/C . Substituting values gives 9.0 × 10⁴ N/C, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency. This aligns with NCERT Class 11 treatment, emphasizing conservation, symmetry and dimensional consistency useful for CBSE, NEET and CUET.

Ref: NCERT > Physics Book > Electric Charges and Fields > Continuous Charge Distribution

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Continuous Charge Distribution

An infinite line charge has \( \lambda = 9 \times 10^{-7} \, \text{C/m} \). What is the electric field at 30 cm?

**Charge density formulation** allows integration over extended bodies, but highly symmetric cases yield simple expressions. Infinite line gives E ∝ λ/r, unlike point charge 1/r², reflecting different geometry of source. E = (2 k λ/r) , k = 9 × 10⁹ N·m²/C² . E = (2 × 9 × 10⁹ × 9 × 10⁻⁷/0.3) = 5.4 × 10⁴ N/C . Substituting values gives 5.4 × 10⁴ N/C, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

Ref: NCERT > Physics Book > Electric Charges and Fields > Continuous Charge Distribution

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Continuous Charge Distribution

An infinite line charge has \( \lambda = 7 \times 10^{-7} \, \text{C/m} \). What is the electric field at 25 cm?

**Charge density formulation** allows integration over extended bodies, but highly symmetric cases yield simple expressions. Infinite line gives E ∝ λ/r, unlike point charge 1/r², reflecting different geometry of source. E = (2 k λ/r) , k = 9 × 10⁹ N·m²/C² . E = (2 × 9 × 10⁹ × 7 × 10⁻⁷/0.25) = 5.04 × 10⁴ N/C . Substituting values gives 5.04 × 10⁴ N/C, which matches expected magnitude for this electrostatic configuration, confirming Coulomb's and Gauss's principles and charge quantization consistency.

Ref: NCERT > Physics Book > Electric Charges and Fields > Continuous Charge Distribution

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