Galvanometer half-deflection experiment

A galvanometer can help measure its own resistance. Watch its pointer, add a parallel path, then restore a precise fraction of the original deflection. The trap is that adding the path also changes the battery current. A successful experiment keeps track of both changes.

The procedure joins three ideas already familiar from circuits: equal voltage across parallel branches, conservation of current at a junction, and a source whose current depends on its load.

What you will learn to do

Distinguish coil current from total source current.
Derive the finite-series-resistance half-deflection relation.
Infer the figure of merit with consistent scale units.
Evaluate repeat observations and the limits of the shunt-equals-coil approximation.

Useful before this chapterElectrical instruments: meter ranges and bridge balances →Kirchhoff networks and current direction →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Galvanometer half-deflection experiment
Learning goals:
- Distinguish coil current from total source current.
- Derive the finite-series-resistance half-deflection relation.
- Infer the figure of merit with consistent scale units.
- Evaluate repeat observations and the limits of the shunt-equals-coil approximation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

BEFORE YOU READ

Bring back one idea

Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.

If the pointer obeys I = kθ, what does half the original deflection tell you about the coil current?

Compare your reasoning

With the same calibration and a zero-corrected deflection in the linear range, the coil current is halved. It says nothing by itself about the current through the battery, because the added shunt provides another path.

Revisit the explanation →

01 / Galvanometer half-deflection experiment

The pointer measures coil current, not every current in the circuit

In a moving-coil galvanometer with a radial magnetic field, the current produces a torque proportional to coil current. A restoring suspension supplies an opposing torque proportional to angular deflection. At equilibrium their balance makes deflection proportional to current over the usable linear range. A pointer scale can express that deflection in divisions instead of radians, with a corresponding calibration constant.

Write I_g = kθ, where k is the figure of merit in amperes per scale division. A smaller k gives more deflection for the same small current. The current sensitivity θ/I_g is the reciprocal of k. Neither quantity is the resistance G of the coil. We will first infer G from a deflection ratio and then use the source emf to infer k.

I_g=k\theta

Use the same scale unit for all deflections. Zero the pointer mechanically before comparing readings; an unremoved pointer offset would spoil the current ratio.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Galvanometer half-deflection experiment
Focus: The pointer measures coil current, not every current in the circuit
Learning goals:
- Distinguish coil current from total source current.
- Derive the finite-series-resistance half-deflection relation.
- Infer the figure of merit with consistent scale units.
- Evaluate repeat observations and the limits of the shunt-equals-coil approximation.
Equations in this section (LaTeX):
I_g=k\theta
Meaning: Use the same scale unit for all deflections. Zero the pointer mechanically before comparing readings; an unremoved pointer offset would spoil the current ratio.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

02 / Galvanometer half-deflection experiment

Keep the shunt across the coil and the limiting resistor in series

The source E and limiting resistance R feed the coil G. Closing K connects adjustable shunt S across the same two coil terminals. R remains in the shared source path. The drawing is a connection diagram, not a suggested physical layout.

Start with the shunt branch open and a large series resistance. Close the source circuit briefly, adjust to a clear deflection within range, and record R and θ₀. Keep that series setting and the source emf unchanged. Connect the shunt and adjust S until the coil deflection is θ₀/2. Record S, then open the source circuit between observations to limit heating and drift. Use the instrument within its rated current range.

R denotes all fixed resistance in the shared series path outside the galvanometer in our model. If the source has known internal resistance r and the resistance box is R_box, use R = R_box+r. In the numerical examples, internal resistance and lead/contact resistance are explicitly neglected. An unknown changing source resistance is an experimental limitation, not permission to insert the box value blindly.

The shunt must connect to the same two nodes as the coil. Putting it in series can also reduce deflection, but it is a different experiment with a different equation. Record the connections as carefully as the resistance-box numbers.

STOP & THINK

Closing a positive-resistance shunt across the coil, with E and R fixed, does what to the total source current?

It stays exactly fixed.
It increases.
It halves with the coil current.

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Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Galvanometer half-deflection experiment
Focus: Keep the shunt across the coil and the limiting resistor in series
Learning goals:
- Distinguish coil current from total source current.
- Derive the finite-series-resistance half-deflection relation.
- Infer the figure of merit with consistent scale units.
- Evaluate repeat observations and the limits of the shunt-equals-coil approximation.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

03 / Galvanometer half-deflection experiment

Use the two circuits before imposing half deflection

With the shunt open, the same current passes through R and G. With the shunt closed, let i be the new coil current and i_s the shunt current. Equal terminal voltages give iG = i_sS. The current in the series resistor is now i+i_s, not i. Applying the source loop relation therefore gives E = R(i+i_s)+iG.

Substitute i_s = iG/S, collect the terms multiplying i, and compare with the original coil current. The source emf cancels from the ratio only because it is unchanged between the two measurements. The resistance formula does not require measuring E; determining current per division later does.

\begin{aligned} I_0&=\frac{E}{R+G}\\ i&=\frac{ES}{RG+RS+GS}\\ \frac{i}{I_0}&=\frac{S(R+G)}{RG+RS+GS} \end{aligned}

These equations retain the change in total current when S is connected. All resistances are positive and their temperature-dependent changes are neglected.

\frac{i}{I_0}=\frac12\quad\Longrightarrow\quad G=\frac{RS}{R-S}

Cross-multiplication gives 2S(R+G)=RG+RS+GS, hence G(R−S)=RS.

Explain this equation

Read it in words

Coil resistance equals the product of series and shunt resistances divided by their difference.

Symbols and units

$G$: Galvanometer coil resistanceΩ
$R$: All fixed shared series resistance outside the coilΩ
$S$: Shunt resistance at half the original coil deflectionΩ

Signs and operations

R−S must be positive for a positive finite G. The minus sign is a result of current redistribution, not a chosen voltage direction.

When it applies

Same emf and series setting before/after; ohmic resistances; coil deflection proportional to coil current; negligible unmodelled loading.

Make it concrete

R=900 Ω and S=90 Ω give G=100 Ω. Substituting into the two-current equations halves the coil current.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Equation (LaTeX): \frac{i}{I_0}=\frac12\quad\Longrightarrow\quad G=\frac{RS}{R-S}
Read it as: Coil resistance equals the product of series and shunt resistances divided by their difference.
Symbols:
G: Galvanometer coil resistance; Ω
R: All fixed shared series resistance outside the coil; Ω
S: Shunt resistance at half the original coil deflection; Ω
Signs and operations: R−S must be positive for a positive finite G. The minus sign is a result of current redistribution, not a chosen voltage direction.
Conditions: Same emf and series setting before/after; ohmic resistances; coil deflection proportional to coil current; negligible unmodelled loading.

Help me explain the equation in words, predict how changing a quantity affects the result, and apply it to a small numerical example.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Galvanometer half-deflection experiment
Focus: Use the two circuits before imposing half deflection
Learning goals:
- Distinguish coil current from total source current.
- Derive the finite-series-resistance half-deflection relation.
- Infer the figure of merit with consistent scale units.
- Evaluate repeat observations and the limits of the shunt-equals-coil approximation.
Equations in this section (LaTeX):
\begin{aligned} I_0&=\frac{E}{R+G}\\ i&=\frac{ES}{RG+RS+GS}\\ \frac{i}{I_0}&=\frac{S(R+G)}{RG+RS+GS} \end{aligned}
Meaning: These equations retain the change in total current when S is connected. All resistances are positive and their temperature-dependent changes are neglected.
\frac{i}{I_0}=\frac12\quad\Longrightarrow\quad G=\frac{RS}{R-S}
Meaning: Cross-multiplication gives 2S(R+G)=RG+RS+GS, hence G(R−S)=RS.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

04 / Galvanometer half-deflection experiment

Verify a resistance result by reconstructing both currents

WORKED EXAMPLE

A shunt is close to, but not equal to, the coil resistance

At R = 900 Ω, a shunt S = 90 Ω reduces the deflection from 30 to 15 divisions. Find G and check the ratio using a stable 1.50 V source with negligible internal resistance.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Galvanometer half-deflection experiment
Focus: Verify a resistance result by reconstructing both currents
Learning goals:
- Distinguish coil current from total source current.
- Derive the finite-series-resistance half-deflection relation.
- Infer the figure of merit with consistent scale units.
- Evaluate repeat observations and the limits of the shunt-equals-coil approximation.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

05 / Galvanometer half-deflection experiment

Choose R to control the approximation, then check readability

Rewrite the exact result as G = S/(1−S/R). If S/R is small, the denominator is near one and G≈S. The fractional underestimate from using S instead of the exact G is (G−S)/G = S/R. This identifies the small parameter directly.

A large R reduces this approximation error but also reduces the original current and deflection for a given source. A tiny deflection makes a half-reading hard to distinguish. Instrument range, pointer resolution, source stability and heating therefore matter together. There is no useful experiment if the chosen approximation is excellent but the observation is unreadable.

WORKED EXAMPLE

Decide whether an approximation meets a stated tolerance

A trial uses R = 2000 Ω and reaches half deflection at S = 80 Ω. Is using G≈S accurate to within 1% relative to the exact G?

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Galvanometer half-deflection experiment
Focus: Choose R to control the approximation, then check readability
Learning goals:
- Distinguish coil current from total source current.
- Derive the finite-series-resistance half-deflection relation.
- Infer the figure of merit with consistent scale units.
- Evaluate repeat observations and the limits of the shunt-equals-coil approximation.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

06 / Galvanometer half-deflection experiment

Resistance and current-per-division answer different questions

Return to the unshunted state when calculating the figure of merit from the original deflection. Its coil current is E/(R+G). Divide that current by θ₀ in the chosen scale units. Using the total source current from the shunted circuit with the coil deflection would mix different currents.

The resistance measurement uses a ratio, so a stable emf cancels. The figure-of-merit measurement needs an emf or an independently known coil current. If E is the emf, include source internal resistance in R. A measured terminal voltage under load can instead be used with the corresponding external circuit equation, but must not be silently substituted into an emf formula with internal resistance counted again.

k=\frac{E}{(R+G)\theta_0}

The unit is A/division when θ₀ is recorded in scale divisions. A numerical value in μA/division differs by a factor of one million.

WORKED EXAMPLE

Calibrate the pointer after finding G

Use E = 1.50 V, R = 900 Ω, G = 100 Ω and an unshunted deflection of 30 divisions. Find k in μA per division.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Galvanometer half-deflection experiment
Focus: Resistance and current-per-division answer different questions
Learning goals:
- Distinguish coil current from total source current.
- Derive the finite-series-resistance half-deflection relation.
- Infer the figure of merit with consistent scale units.
- Evaluate repeat observations and the limits of the shunt-equals-coil approximation.
Equations in this section (LaTeX):
k=\frac{E}{(R+G)\theta_0}
Meaning: The unit is A/division when θ₀ is recorded in scale divisions. A numerical value in μA/division differs by a factor of one million.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

07 / Galvanometer half-deflection experiment

Repeat with a changed series resistance to test the model

Ideal-model consistency dataset, E = 1.50 V, negligible source resistance. Shunt values in rows 2–3 are rounded to the displayed precision.
R / Ω	Original divisions	Half divisions	S / Ω
900	30	15	90.00
1400	20	10	93.33
1900	15	7.5	95.00

The same coil can require different half-deflection shunts when R changes. That is expected from the exact model, not evidence that the coil resistance has changed. Calculate G independently in each row before comparing. Then infer k from each original deflection. This checks whether the two measurements are consistent with a common coil and calibration.

This illustrative table is generated from a simple circuit, not presented as real laboratory data. Actual observations need resistance-box accuracy, pointer interpolation, stable emf and repeated readings. If the inferred G values drift systematically, investigate heating, contact resistance, pointer offset and nonlinearity before averaging them.

WORKED EXAMPLE

A changed shunt can describe the same coil

For the third row, calculate G and k.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Galvanometer half-deflection experiment
Focus: Repeat with a changed series resistance to test the model
Learning goals:
- Distinguish coil current from total source current.
- Derive the finite-series-resistance half-deflection relation.
- Infer the figure of merit with consistent scale units.
- Evaluate repeat observations and the limits of the shunt-equals-coil approximation.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

08 / Galvanometer half-deflection experiment

A difference in the denominator can magnify uncertainty

The exact expression does not eliminate observational uncertainty. At fixed R, increasing S increases G. At fixed S, increasing R decreases G slightly. These monotonicities let us find hard bounds without assuming every error combines with the same sign. If the allowed interval crosses R=S, the inferred model is poorly constrained or inconsistent; it is not a harmless negative-resistance result.

A half-reading also has uncertainty. With a coarse pointer scale, record the bracket of S values that straddles half deflection instead of inventing an exact setting. The general current-ratio expression can be used when the observed ratio is not exactly one half. Keep the observed deflections in the record so that later analysis can test that assumption.

WORKED EXAMPLE

Bracket the coil resistance from bounded settings

Assume half deflection is exact, R lies from 890 to 910 Ω, and S from 89 to 91 Ω. Bound G.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Galvanometer half-deflection experiment
Focus: A difference in the denominator can magnify uncertainty
Learning goals:
- Distinguish coil current from total source current.
- Derive the finite-series-resistance half-deflection relation.
- Infer the figure of merit with consistent scale units.
- Evaluate repeat observations and the limits of the shunt-equals-coil approximation.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

Half the deflection means half the coil current only in the calibrated linear range.
Adding the shunt changes the equivalent load and therefore the source current.
At half deflection G = RS/(R−S); use G≈S only with a justified large-R approximation.
Figure of merit is current per scale division; current sensitivity is its reciprocal.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1

At fixed R = 600 Ω, a shunt S = 120 Ω gives exactly half the unshunted deflection. Find G in Ω.

Answer in Ω.

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SELF CHECK 2

An unshunted coil G = 100 Ω in series with R = 1900 Ω carries current from E = 2.0 V and deflects 25 divisions. Find k in μA/division; neglect source resistance.

Answer in μA/division.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 3

At half deflection, S/R = 0.025. What percentage does the approximation G≈S underestimate the exact G by?

Answer in %.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 4

The source emf falls between the initial and shunted observations. Which conclusion is justified?

The ratio method is unchanged because emf always cancels.
The original half-deflection formula needs the same-emf assumption rechecked.
Only k changes; G is guaranteed exact.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

APPLY THE CHAPTER

Take the ideas into a problem

Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.

Half the pointer, a changing source currentIntermediate · Free sample · Hints and guided reasoning Calibrate what one division meansBeginner · Hints and guided reasoning Does the shortcut meet the tolerance?Advanced · Hints and guided reasoning

COME BACK TO THE IDEA

What can you explain later?

Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.

Next study session

At half deflection, why is the shunt resistance smaller than the coil resistance?

Compare your reasoning

Adding the shunt lowers the equivalent load and increases the total source current. The coil carries half its original current, so the shunt must carry more than that half. Equal branch voltages then require a smaller shunt resistance.

Revisit the explanation →

A few days later: change the situation

Why can a source-voltage change between the two deflections spoil the resistance measurement?

Compare your reasoning

The derivation cancels a common source emf from the before and after currents. If it changes between readings, a different ratio of emfs remains. A pointer at half scale then need not correspond to the resistance relation derived for a stable source.

Revisit the explanation →

Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.

Back to chapter contents ↑