2025

1. Work to bring a charge to triangle corner

• Triangle side $4\,\text{cm}$4cm. Charges $q_1 = 2\,\text{nC}$q1​=2nC, $q_2 = 4\,\text{nC}$q2​=4nC at bottom corners. Bring $q_3 = 3\,\text{nC}$q3​=3nC to top corner.
• Diagram: equilateral triangle, label charges A, B, C.

2. Electric field zero between two charges

• $+2\,\mu C$+2μC at $x=0$x=0, $+8\,\mu C$+8μC at $x=1\,\text{m}$x=1m. Find point along x-axis where net electric field is zero.

3. Potential on axis of a uniformly charged rod

• Rod length $0.6\,\text{m}$0.6m, total charge $6\,\mu C$6μC, point along axis $0.2\,\text{m}$0.2m from one end.

4. Torque on a dipole in uniform field

• Dipole $p = 3\times10^{-12}\,C·m$p=3×10−12C⋅m in uniform field $E = 2\times10^3\,V/m$E=2×103V/m at angle $30^\circ$30∘ to field.

5. Series capacitors with dielectric

• $C_1 = 4\,\mu F$C1​=4μF, $C_2 = 6\,\mu F$C2​=6μF in series. Dielectric $\kappa=2$κ=2 in $C_1$C1​. Find new equivalent capacitance.

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2024

1. Field at midpoint of line of two charges

• Charges $+3\,\mu C$+3μC at $x=0$x=0, $+6\,\mu C$+6μC at $x=2\,\text{m}$x=2m. Find field at midpoint.

2. Potential difference between two points

• Charges $+2\,\mu C$+2μC at $x=0$x=0, $+3\,\mu C$+3μC at $x=3\,\text{m}$x=3m. Find potential difference between $x=1\,\text{m}$x=1m and $x=2\,\text{m}$x=2m.

3. Gauss law – field outside a cylinder

• Infinite cylinder radius $0.1\,\text{m}$0.1m, linear charge density $\lambda=4\times10^{-6}\,C/m$λ=4×10−6C/m. Find field at $r=0.2\,\text{m}$r=0.2m.

4. Dipole along axis

• Dipole $p=2\times10^{-12}\,C·m$p=2×10−12C⋅m, point along axis $0.05\,\text{m}$0.05m from center.

5. Parallel plate capacitor with dielectric

• Capacitance $C=5\,\mu F$C=5μF, dielectric $\kappa=3$κ=3 partially fills plates.

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2023

1. Electric field at center of square of charges

• Square side $0.5\,\text{m}$0.5m, corner charges $+1, +2, +3, +4\,\mu C$+1,+2,+3,+4μC. Find net field at center.

2. Work to assemble three charges on a line

• Charges $+2, +3, +4\,\mu C$+2,+3,+4μC at positions 0, 1, 2 m. Find work done to assemble.

3. Potential at axis of charged ring

• Ring radius $0.2\,\text{m}$0.2m, $Q = 6\,\mu C$Q=6μC, point along axis $0.1\,\text{m}$0.1m from center.

4. Torque on a dipole

• Dipole $p=1\times10^{-12}\,C·m$p=1×10−12C⋅m, $E=1000\,V/m$E=1000V/m, angle $45^\circ$45∘.

5. Capacitor in series

• Capacitors $C_1=2\,\mu F$C1​=2μF, $C_2=3\,\mu F$C2​=3μF in series. Find equivalent capacitance.

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2022

1. Electric field due to semicircular rod

• Rod radius $0.1\,\text{m}$0.1m, total charge $Q=4\,\mu C$Q=4μC. Find field at center of circle.

2. Point where field is zero between two charges

• Charges $+5\,\mu C$+5μC and $+20\,\mu C$+20μC separated by $2\,\text{m}$2m. Find point along line of charges where field is zero.

3. Potential at a point due to line charge

• Rod length $0.5\,\text{m}$0.5m, $Q=5\,\mu C$Q=5μC, point along axis $0.1\,\text{m}$0.1m from one end.

4. Torque on a dipole

• Dipole $p=2\times10^{-12}\,C·m$p=2×10−12C⋅m, $E=2000\,V/m$E=2000V/m, angle $60^\circ$60∘.

5. Capacitor partially filled with dielectric

• $C=6\,\mu F$C=6μF, dielectric $\kappa=2$κ=2 fills half the space.

2021

1. Electric field at midpoint of two charges

• Charges $+4\,\mu C$+4μC at $x=0$x=0, $+6\,\mu C$+6μC at $x=1.5\,m$x=1.5m. Find electric field at midpoint.

2. Work to assemble charges on a line

• Charges $+3, +5, +2\,\mu C$+3,+5,+2μC at 0, 1, 2 m. Find total work done.

3. Potential at center of square of charges

• Square side $0.4\,\text{m}$0.4m, corner charges $+2, +2, +2, +2\,\mu C$+2,+2,+2,+2μC. Find potential at center.

4. Torque on a dipole in uniform field

• Dipole $p = 2\times10^{-12}\,C·m$p=2×10−12C⋅m, $E=1500\,V/m$E=1500V/m, $\theta=45^\circ$θ=45∘.

5. Capacitor with dielectric slab

• Parallel plate capacitor $C=4\,\mu F$C=4μF, dielectric $\kappa=3$κ=3 fills half distance. Find new capacitance.

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2020

1. Field at a point on the axis of a charged ring

• Ring radius $0.15\,\text{m}$0.15m, total charge $Q=5\,\mu C$Q=5μC, point $0.1\,\text{m}$0.1m above center.

2. Point where field is zero between unequal charges

• Charges $+3\,\mu C$+3μC at $x=0$x=0, $+12\,\mu C$+12μC at $x=2\,\text{m}$x=2m.

3. Potential due to uniform line charge

• Rod length $0.5\,\text{m}$0.5m, charge $4\,\mu C$4μC, point $0.2\,\text{m}$0.2m from one end along axis.

4. Torque on a dipole in uniform field

• Dipole $p = 1.5\times10^{-12}\,C·m$p=1.5×10−12C⋅m, $E=2000\,V/m$E=2000V/m, angle $60^\circ$60∘.

5. Series capacitors with dielectric in one

• $C_1 = 3\,\mu F$C1​=3μF, $C_2 = 6\,\mu F$C2​=6μF, dielectric $\kappa=2$κ=2 in $C_1$C1​.

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2019

1. Electric field at center of semicircular ring

• Ring radius $0.1\,\text{m}$0.1m, charge $Q=3\,\mu C$Q=3μC.

2. Work done to bring a charge to midpoint of two charges

• Charges $+2, +4\,\mu C$+2,+4μC at ends of 1 m line, bring $q=1\,\mu C$q=1μC to midpoint.

3. Potential on axis of charged disk

• Disk radius $0.2\,\text{m}$0.2m, charge $Q=5\,\mu C$Q=5μC, point $0.1\,\text{m}$0.1m above center.

4. Torque on dipole

• Dipole $p=1\times10^{-12}\,C·m$p=1×10−12C⋅m, $E=1000\,V/m$E=1000V/m, angle $30^\circ$30∘.

5. Capacitor partially filled with dielectric

• $C=5\,\mu F$C=5μF, dielectric $\kappa=3$κ=3 fills half plate distance.

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2018

1. Point where field is zero along line of two charges

• Charges $+4, +16\,\mu C$+4,+16μC separated by 2 m.

2. Electric field at center of square with four charges

• Square side $0.3\,m$0.3m, corner charges $+1, +2, +3, +4\,\mu C$+1,+2,+3,+4μC.

3. Potential at midpoint between two charges

• Charges $+3, +5\,\mu C$+3,+5μC separated by 1 m.

4. Dipole in uniform field

• Dipole $p=2\times10^{-12}\,C·m$p=2×10−12C⋅m, $E=1500\,V/m$E=1500V/m, $\theta=60^\circ$θ=60∘.

5. Series capacitors with dielectric in one

• $C_1=2\,\mu F, C_2=4\,\mu F$C1​=2μF,C2​=4μF, dielectric $\kappa=2$κ=2 in $C_1$C1​.

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2017

1. Field at axis of a charged ring

• Ring radius $0.1\,\text{m}$0.1m, charge $Q=4\,\mu C$Q=4μC, point $0.05\,\text{m}$0.05m above center.

2. Electric field zero between unequal charges

• Charges $+2, +8\,\mu C$+2,+8μC separated by 1 m.

3. Potential on axis of rod

• Rod length 0.5 m, charge 3 μC, point 0.1 m from end.

4. Torque on dipole

• Dipole $p=1\times10^{-12}\,C·m$p=1×10−12C⋅m, $E=2000\,V/m$E=2000V/m, angle $45^\circ$45∘.

5. Capacitor partially filled with dielectric

• $C=3\,\mu F$C=3μF, dielectric $\kappa=2$κ=2 fills half distance.

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2016

1. Work to bring a charge to triangle corner

• Equilateral triangle side 3 cm, charges $q_1=1\,\mu C$q1​=1μC, $q_2=2\,\mu C$q2​=2μC, bring $q_3=3\,\mu C$q3​=3μC to third corner.

2. Electric field at midpoint of two charges

• Charges $+3, +6\,\mu C$+3,+6μC separated by 1 m.

3. Potential at axis of rod

• Rod length 0.6 m, charge 5 μC, point 0.2 m from one end.

4. Dipole in uniform field

• $p=2\times10^{-12}\,C·m$p=2×10−12C⋅m, $E=1000\,V/m$E=1000V/m, angle 30°.

5. Series capacitors with dielectric in one

• $C_1=4\,\mu F, C_2=6\,\mu F$C1​=4μF,C2​=6μF, $\kappa=2$κ=2 in $C_1$C1​.

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2015

1. Electric field at center of square

• Square side 0.4 m, charges $+1, +1, +1, +1\,\mu C$+1,+1,+1,+1μC.

2. Work to assemble 3 charges on line

• Charges $+2, +3, +4\,\mu C$+2,+3,+4μC at 0,1,2 m.

3. Potential at midpoint between two charges

• Charges $+3, +5\,\mu C$+3,+5μC separated by 1 m.

4. Torque on dipole

• $p=1.5\times10^{-12}\,C·m$p=1.5×10−12C⋅m, $E=1500\,V/m$E=1500V/m, angle 45°.

5. Capacitor partially filled with dielectric

• $C=4\,\mu F$C=4μF, dielectric $\kappa=3$κ=3 fills half distance.

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2014

1. Point where field is zero between two charges

• Charges $+3, +12\,\mu C$+3,+12μC separated by 2 m.

2. Field at center of semicircular rod

• Rod radius 0.1 m, charge 4 μC.

3. Potential at axis of charged ring

• Ring radius 0.15 m, charge 5 μC, point 0.1 m above center.

4. Torque on dipole

• $p=2\times10^{-12}\,C·m$p=2×10−12C⋅m, $E=1000\,V/m$E=1000V/m, angle 60°.

5. Capacitors in series

• $C_1=2 μF, C_2=3 μF$C1​=2μF,C2​=3μF, series combination.

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2013

1. Work to bring charge to triangle corner

• Triangle side 3 cm, charges $q_1=1 μC, q_2=2 μC$q1​=1μC,q2​=2μC, bring $q_3=3 μC$q3​=3μC.

2. Electric field zero along line

• Charges $+2, +8 μC$+2,+8μC separated by 1 m.

3. Potential at axis of rod

• Rod length 0.5 m, charge 3 μC, point 0.2 m from end.

4. Dipole in uniform field

• $p=1×10^{-12} C·m$p=1×10−12C⋅m, $E=2000 V/m$E=2000V/m, angle 45°.

5. Capacitor with dielectric

• $C=3 μF$C=3μF, dielectric κ=2 fills half distance.

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2012

1. Electric field at midpoint of two charges

• Charges +3, +6 μC, separation 1 m.

2. Work to assemble three charges on line

• Charges +2, +3, +4 μC at 0,1,2 m.

3. Potential at center of square

• Square side 0.4 m, charges +1, +2, +3, +4 μC.

4. Torque on dipole

• Dipole p=2×10⁻¹² C·m, E=1000 V/m, angle 30°.

5. Series capacitors with dielectric

• C₁=4 μF, C₂=6 μF, dielectric κ=2 in C₁.

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2011

1. Point where field is zero between unequal charges

• Charges +2, +8 μC separated by 1 m.

2. Field at axis of charged ring

• Ring radius 0.15 m, charge 5 μC, point 0.1 m above center.

3. Potential on axis of rod

• Rod length 0.6 m, charge 5 μC, point 0.2 m from one end.

4. Torque on dipole

• Dipole p=1×10⁻¹² C·m, E=1500 V/m, angle 45°.

5. Capacitor partially filled with dielectric

• C=4 μF, dielectric κ=3 fills half distance.

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2010

1. Work to bring a charge to triangle corner

• Triangle side 3 cm, charges q₁=1 μC, q₂=2 μC, bring q₃=3 μC.

2. Electric field at midpoint of two charges

• Charges +3, +6 μC, separation 1 m.

3. Potential at center of square of charges

• Square side 0.4 m, charges +1, +1, +1, +1 μC.

4. Dipole in uniform field

• Dipole p=1×10⁻¹² C·m, E=1000 V/m, angle 30°.

5. Capacitors in series

• C₁=2 μF, C₂=3 μF, series combination.

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Electrostatics Solutions – 2025

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Q1: Work to bring a charge to the corner of a triangle

Problem:

• Equilateral triangle, side $4\,\text{cm}$4cm
• Charges $q_1=2\,\text{nC}$q1​=2nC, $q_2=4\,\text{nC}$q2​=4nC at bottom corners
• Bring $q_3=3\,\text{nC}$q3​=3nC to top corner

Diagram description:

• Draw triangle ABC, bottom corners A ($q_1$q1​) and B ($q_2$q2​), top corner C ($q_3$q3​ moves here)

Solution:

1. Distance between charges: $r = 0.04\,\text{m}$r=0.04m
2. Potential at C due to $q_1$q1​ and $q_2$q2​:

$$V_C = k \left(\frac{q_1}{r} + \frac{q_2}{r}\right) = 9\times10^9 \left(\frac{2\times10^{-9}}{0.04} + \frac{4\times10^{-9}}{0.04}\right) = 9\times10^9 \cdot 1.5\times10^{-7} = 1350\,\text{V}$$VC​=k(rq1​​+rq2​​)=9×109(0.042×10−9​+0.044×10−9​)=9×109⋅1.5×10−7=1350V

3. Work done to bring $q_3$q3​ from infinity:

$$W = q_3 V_C = 3\times10^{-9} \cdot 1350 = 4.05\times10^{-6}\,J$$W=q3​VC​=3×10−9⋅1350=4.05×10−6J

✅ Answer: $W = 4.05\,\mu J$W=4.05μJ

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Q2: Electric field zero between two charges

Problem:

• Charges $+2\,\mu C$+2μC at $x=0$x=0, $+8\,\mu C$+8μC at $x=1\,m$x=1m
• Find point along x-axis where net field is zero

Diagram description:

• Draw x-axis, charge +2 at 0, +8 at 1 m, point P somewhere between

Solution:

1. Let point P be at distance $x$x from smaller charge (+2 μC).

• Electric field due to +2 μC: $E_1 = k \frac{2}{x^2}$E1​=kx22​
• Electric field due to +8 μC: $E_2 = k \frac{8}{(1-x)^2}$E2​=k(1−x)28​

2. Set $E_1 = E_2$E1​=E2​ (opposite directions):

$$\frac{2}{x^2} = \frac{8}{(1-x)^2} \implies \frac{1}{x} = \frac{2}{1-x} \implies 1-x = 2x \implies x = \frac{1}{3}\,m$$x22​=(1−x)28​⟹x1​=1−x2​⟹1−x=2x⟹x=31​m

✅ Answer: $x = 0.333\,m$x=0.333m from +2 μC

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Q3: Potential on axis of a uniformly charged rod

Problem:

• Rod length $0.6\,\text{m}$0.6m, total charge $Q=6\,\mu C$Q=6μC
• Point along axis $0.2\,\text{m}$0.2m from one end

Diagram description:

• Horizontal rod along x-axis, left end at origin, point P at x=0.2 m

Solution:

1. Linear charge density: $\lambda = Q/L = 6\times10^{-6}/0.6 = 1\times10^{-5}\,C/m$λ=Q/L=6×10−6/0.6=1×10−5C/m
2. Potential at point along axis:

$$V = k \int_0^L \frac{\lambda dx}{r} = 9\times10^9 \cdot 10^{-5} \int_0^{0.6} \frac{dx}{0.2 + x}$$V=k∫0L​rλdx​=9×109⋅10−5∫00.6​0.2+xdx​

3. Solve integral:

$$V = 9\times10^4 \ln\frac{0.2+0.6}{0.2} = 9\times10^4 \ln 4 \approx 9\times10^4 \cdot 1.386 = 1.247\times10^5\,V$$V=9×104ln0.20.2+0.6​=9×104ln4≈9×104⋅1.386=1.247×105V

✅ Answer: $V \approx 1.25\times10^5\,V$V≈1.25×105V

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Q4: Torque on a dipole in uniform field

Problem:

• Dipole $p = 3\times10^{-12}\,C·m$p=3×10−12C⋅m
• Uniform field $E = 2\times10^3\,V/m$E=2×103V/m, angle $\theta = 30^\circ$θ=30∘

Diagram description:

• Draw dipole vector at 30° to field vector

Solution:

1. Torque formula:

$$\tau = p E \sin\theta$$τ=pEsinθ

2. Substituting values:

$$\tau = 3\times10^{-12} \cdot 2\times10^3 \cdot \sin30^\circ = 3\times10^{-12} \cdot 2\times10^3 \cdot 0.5 = 3\times10^{-9}\,N·m$$τ=3×10−12⋅2×103⋅sin30∘=3×10−12⋅2×103⋅0.5=3×10−9N⋅m

✅ Answer: $\tau = 3\times10^{-9}\,N·m$τ=3×10−9N⋅m

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Q5: Series capacitors with dielectric

Problem:

• Capacitors in series: $C_1=4\,\mu F$C1​=4μF, $C_2=6\,\mu F$C2​=6μF
• Dielectric $\kappa = 2$κ=2 inserted in $C_1$C1​

Diagram description:

• Two capacitors in series, show dielectric slab in C₁

Solution:

1. Capacitance of C₁ after dielectric:

$$C_1′ = \kappa C_1 = 2 \cdot 4 = 8\,\mu F$$C1′​=κC1​=2⋅4=8μF

2. Equivalent capacitance for series combination:

$$\frac{1}{C_{eq}} = \frac{1}{C_1′} + \frac{1}{C_2} = \frac{1}{8} + \frac{1}{6} = \frac{7}{24} \implies C_{eq} = \frac{24}{7} \approx 3.43\,\mu F$$Ceq​1​=C1′​1​+C2​1​=81​+61​=247​⟹Ceq​=724​≈3.43μF

✅ Answer: $C_{eq} \approx 3.43\,\mu F$Ceq​≈3.43μF

Electrostatics Solutions – 2025

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Q1: Work to bring a charge to the corner of a triangle

Problem:

• Equilateral triangle, side $4\,\text{cm}$4cm
• Charges $q_1=2\,\text{nC}$q1​=2nC, $q_2=4\,\text{nC}$q2​=4nC at bottom corners
• Bring $q_3=3\,\text{nC}$q3​=3nC to top corner

Diagram description:

• Draw triangle ABC, bottom corners A ($q_1$q1​) and B ($q_2$q2​), top corner C ($q_3$q3​ moves here)

Solution:

1. Distance between charges: $r = 0.04\,\text{m}$r=0.04m
2. Potential at C due to $q_1$q1​ and $q_2$q2​:

$$V_C = k \left(\frac{q_1}{r} + \frac{q_2}{r}\right) = 9\times10^9 \left(\frac{2\times10^{-9}}{0.04} + \frac{4\times10^{-9}}{0.04}\right) = 9\times10^9 \cdot 1.5\times10^{-7} = 1350\,\text{V}$$VC​=k(rq1​​+rq2​​)=9×109(0.042×10−9​+0.044×10−9​)=9×109⋅1.5×10−7=1350V

3. Work done to bring $q_3$q3​ from infinity:

$$W = q_3 V_C = 3\times10^{-9} \cdot 1350 = 4.05\times10^{-6}\,J$$W=q3​VC​=3×10−9⋅1350=4.05×10−6J

✅ Answer: $W = 4.05\,\mu J$W=4.05μJ

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Q2: Electric field zero between two charges

Problem:

• Charges $+2\,\mu C$+2μC at $x=0$x=0, $+8\,\mu C$+8μC at $x=1\,m$x=1m
• Find point along x-axis where net field is zero

Diagram description:

• Draw x-axis, charge +2 at 0, +8 at 1 m, point P somewhere between

Solution:

1. Let point P be at distance $x$x from smaller charge (+2 μC).

• Electric field due to +2 μC: $E_1 = k \frac{2}{x^2}$E1​=kx22​
• Electric field due to +8 μC: $E_2 = k \frac{8}{(1-x)^2}$E2​=k(1−x)28​

2. Set $E_1 = E_2$E1​=E2​ (opposite directions):

$$\frac{2}{x^2} = \frac{8}{(1-x)^2} \implies \frac{1}{x} = \frac{2}{1-x} \implies 1-x = 2x \implies x = \frac{1}{3}\,m$$x22​=(1−x)28​⟹x1​=1−x2​⟹1−x=2x⟹x=31​m

✅ Answer: $x = 0.333\,m$x=0.333m from +2 μC

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Q3: Potential on axis of a uniformly charged rod

Problem:

• Rod length $0.6\,\text{m}$0.6m, total charge $Q=6\,\mu C$Q=6μC
• Point along axis $0.2\,\text{m}$0.2m from one end

Diagram description:

• Horizontal rod along x-axis, left end at origin, point P at x=0.2 m

Solution:

1. Linear charge density: $\lambda = Q/L = 6\times10^{-6}/0.6 = 1\times10^{-5}\,C/m$λ=Q/L=6×10−6/0.6=1×10−5C/m
2. Potential at point along axis:

$$V = k \int_0^L \frac{\lambda dx}{r} = 9\times10^9 \cdot 10^{-5} \int_0^{0.6} \frac{dx}{0.2 + x}$$V=k∫0L​rλdx​=9×109⋅10−5∫00.6​0.2+xdx​

3. Solve integral:

$$V = 9\times10^4 \ln\frac{0.2+0.6}{0.2} = 9\times10^4 \ln 4 \approx 9\times10^4 \cdot 1.386 = 1.247\times10^5\,V$$V=9×104ln0.20.2+0.6​=9×104ln4≈9×104⋅1.386=1.247×105V

✅ Answer: $V \approx 1.25\times10^5\,V$V≈1.25×105V

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Q4: Torque on a dipole in uniform field

Problem:

• Dipole $p = 3\times10^{-12}\,C·m$p=3×10−12C⋅m
• Uniform field $E = 2\times10^3\,V/m$E=2×103V/m, angle $\theta = 30^\circ$θ=30∘

Diagram description:

• Draw dipole vector at 30° to field vector

Solution:

1. Torque formula:

$$\tau = p E \sin\theta$$τ=pEsinθ

2. Substituting values:

$$\tau = 3\times10^{-12} \cdot 2\times10^3 \cdot \sin30^\circ = 3\times10^{-12} \cdot 2\times10^3 \cdot 0.5 = 3\times10^{-9}\,N·m$$τ=3×10−12⋅2×103⋅sin30∘=3×10−12⋅2×103⋅0.5=3×10−9N⋅m

✅ Answer: $\tau = 3\times10^{-9}\,N·m$τ=3×10−9N⋅m

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Q5: Series capacitors with dielectric

Problem:

• Capacitors in series: $C_1=4\,\mu F$C1​=4μF, $C_2=6\,\mu F$C2​=6μF
• Dielectric $\kappa = 2$κ=2 inserted in $C_1$C1​

Diagram description:

• Two capacitors in series, show dielectric slab in C₁

Solution:

1. Capacitance of C₁ after dielectric:

$$C_1′ = \kappa C_1 = 2 \cdot 4 = 8\,\mu F$$C1′​=κC1​=2⋅4=8μF

2. Equivalent capacitance for series combination:

$$\frac{1}{C_{eq}} = \frac{1}{C_1′} + \frac{1}{C_2} = \frac{1}{8} + \frac{1}{6} = \frac{7}{24} \implies C_{eq} = \frac{24}{7} \approx 3.43\,\mu F$$Ceq​1​=C1′​1​+C2​1​=81​+61​=247​⟹Ceq​=724​≈3.43μF

✅ Answer: $C_{eq} \approx 3.43\,\mu F$Ceq​≈3.43μF