Current Electricity

Step 1: Kinetic energy of charged particles in a magnetic field.
The energy of a charged particle moving in a magnetic field is given by: where is the mass and is the velocity.
Step 2: Relationship between mass and energy.
Since the mass of the particle is 4 times the mass of the proton, the velocity of the particle will be slower, but the energy is proportional to the mass. Thus, the energy of the proton will be four times that of the particle.
Step 3: Conclusion.
The energy of the proton will be , so the correct answer is (C).

Step 1: Identify galvanometer parameters.
Shunt resistance:

Step 2: Voltage across galvanometer at full scale deflection.


Step 3: Current through shunt resistance.


Step 4: Total current required for full scale deflection.

Step 1: Using the potentiometer principle.
The potentiometer works on the principle of comparing potential differences. The ratio of the lengths at the null points is proportional to the ratio of the potential differences.
Step 2: Applying the given data.
For the first case, the null point is at 0.9 m, and for the second case, the null point is at 0.3 m. Hence, the ratio of the potential differences is: Step 3: Conclusion.
The correct answer is (C), 2:3.

Step 1: Define sensitivity of potentiometer.
Sensitivity of a potentiometer is defined as its ability to detect small changes in potential difference.

Step 2: Relation with potential gradient.
Sensitivity is inversely proportional to the potential gradient along the wire.


Step 3: Analyze given options.
Increasing the potential gradient reduces the sensitivity of the potentiometer.

Step 4: Conclusion.
Sensitivity can be decreased by increasing the potential gradient along the wire.

Step 1: Understanding Kirchhoff's Current Law (KCL).
Kirchhoff's current law states that the algebraic sum of currents at a junction is zero.
This law is based on the conservation of electric charge.

Step 2: Understanding Kirchhoff's Voltage Law (KVL).
Kirchhoff's voltage law states that the algebraic sum of all potential differences in a closed loop is zero.
This law is based on the conservation of energy.

Step 3: Conclusion.
KCL is based on conservation of charge and KVL is based on conservation of energy.

Step 1: Understanding the meter bridge principle.
In a meter bridge, the null point is the point where the ratio of resistances on both sides of the bridge is equal: where and are the resistances in the two gaps, and and are the distances from the left and right ends of the bridge, respectively. Step 2: Apply the conditions.
Initially, the null point is at 40 cm, so: After connecting the 30 resistor in series with the smaller resistance, the null point shifts to 60 cm, so: Step 3: Solving the system of equations.
From the first equation: Substitute this into the second equation: Solving for : Multiply through by 3: Simplifying: Step 4: Conclusion.
Thus, the value of the smaller resistance is 24 , which corresponds to option (B).

Step 1: Understanding the problem.
An ammeter is created by shunting a galvanometer. The total current through the circuit is divided between the galvanometer and the shunt. The resistance of the shunt controls the range of the ammeter. In this case, the galvanometer is shunted with resistance , and the range is to be doubled. Step 2: Relationship for doubling the range.
For a given galvanometer with a shunt resistance , the total current passing through the circuit is divided between the galvanometer and the shunt. To double the range, the additional shunt resistance needed is derived from the formula: Step 3: Conclusion.
Therefore, the additional shunt required to double the range is , which corresponds to option (A).

Step 1: Understanding the relationship.
The sensitivity of a moving coil galvanometer is given by: Where: - is the number of turns in the coil, - is the magnetic induction, - is the twist constant of the suspension. Thus, the sensitivity is inversely proportional to the number of turns . Step 2: Conclusion.
The sensitivity is inversely proportional to the number of turns in the coil, so the correct answer is (B) number of turns in the coil.

Step 1: Understanding the relation.
In a potentiometer, the balancing length is directly proportional to the length of the wire. If the length of the wire is doubled, the balancing length will also double. Step 2: Calculation of new balancing length.
Initially, the balancing length is 1.5 m. If the length of the wire is doubled, the new balancing length will be: Thus, the correct answer is (D) 3 m.

Step 1: Understanding the current distribution.
The sum of currents entering a junction is equal to the sum of currents leaving the junction (Kirchhoff's current law). Given: - The current entering the junction is . - The current leaving the junction is . To balance the current, we have: Thus, the correct answer is (A) 0.7 A.

Step 1: Understanding potentiometer setup.
The internal resistance of the cell can be determined by the change in the balancing length when the cell is shunted with resistance . The relation between the lengths before and after shunting is:

Step 2: Solving for .
Since and are in the same ratio, we deduce that .

Step 3: Conclusion.
The internal resistance of the cell is .

Step 1: Current distribution.
Let total current be .
Current through ammeter coil:
Current through shunt:

Step 2: Voltage equality condition.

Step 3: Substituting values.

Step 4: Conclusion.
The shunt resistance is .

Step 1: Understanding the issue with contact resistance.
Contact resistance in a meter bridge can cause errors in measurements of resistance. To minimize this error, it is important to eliminate or reduce the effect of contact resistance.

Step 2: Analyzing the options.
- (A) Using a non-uniform meter bridge wire will introduce further errors, as the resistance along the wire will not be consistent. - (B) Repeating the experiment by interchanging the resistance in the gaps helps to minimize errors due to contact resistance because the contact resistance will affect both resistances equally, thus canceling out any error. - (C) Increasing the current flowing through the meter bridge wire may increase the effect of contact resistance. - (D) Changing the value of the known resistance in the gap does not eliminate contact resistance errors.

Step 3: Conclusion.
The best method to minimize errors due to contact resistance is to repeat the experiment by interchanging the resistances in the gaps. Thus, the correct answer is (B).

Step 1: Apply Kirchhoff’s current law at junction.
Sum of currents entering a junction equals sum of currents leaving the junction.

Step 2: Use given current directions and values.
By applying KCL at appropriate nodes and considering sign convention, we obtain:

Step 3: Interpretation of negative sign.
Negative sign indicates actual direction of current is opposite to the assumed direction.

Step 1: Use potentiometer principle.
The emf of the cell is proportional to the balancing length:

Step 2: Write relation for original wire.


Step 3: Find new potential gradient.
New length of wire New potential gradient

Step 4: Find new balancing length.

Step 1: Relation between emf and balancing length.
In a potentiometer, emf is directly proportional to balancing length: Step 2: Effect of shunting the cell.
When the cell is shunted with resistance equal to its internal resistance, the effective emf becomes half of its original value.
Step 3: Find new balancing length.

Step 1: Understanding the problem.
To extend the range of a galvanometer to measure higher voltages, a series resistance must be added. The total voltage across the galvanometer plus the series resistance is equal to the voltage we wish to measure. Using the concept of current, the current through the galvanometer when the maximum voltage is applied is . To measure volts, the total resistance must be such that the current remains the same.
Step 2: Using the formula.
The total resistance required to measure volts is , where is the resistance of the galvanometer and is the maximum voltage for which the galvanometer is calibrated.
Step 3: Conclusion.
Thus, the correct answer is (B) .

Step 1: Identify components of the circuit.
The circuit contains two cells of emf and connected in opposition and a resistor of .
Step 2: Find net emf.
Since the cells oppose each other,
Step 3: Apply Kirchhoff’s loop law.

Step 4: Calculate current.

Step 5: Conclusion.
The current flowing through the circuit is .

Step 1: Resistance of a coil.
Resistance of a wire is
Step 2: Relation with number of turns.
Total length of wire is proportional to number of turns.
Step 3: Effect of decreasing turns.
When number of turns decreases, the length of wire decreases.
Step 4: Conclusion.
Hence, resistance of galvanometer decreases.

Step 1: Use potentiometer principle.
Balancing length is directly proportional to the net e.m.f.
Step 2: Write equations for both cases.
When cells are connected in series aiding: When polarity of is reversed:
Step 3: Convert proportionality into equations.

Step 4: Solve the equation.

Step 5: Conclusion.
The ratio .

Step 1: Understanding Sensitivity.
Sensitivity of a galvanometer is defined as the amount of current required to produce a unit deflection. The lower the current required, the more sensitive the galvanometer is.
Step 2: Comparing the Galvanometers.
- G1 requires 2 mA to produce a deflection. - G2 requires 3 mA to produce the same deflection. Since G1 requires a smaller current, it is more sensitive.
Step 3: Conclusion.
Therefore, G1 is more sensitive than G2, which makes option (B) the correct answer.

Step 1: Formula for resistance.
The resistance of a wire is given by the formula: where is the resistivity, is the length, and is the cross-sectional area of the wire. The area is related to the diameter by: Step 2: Minimizing resistance.
For minimum resistance, we need to decrease the length and increase the diameter . This will maximize the area and minimize the resistance.
Step 3: Conclusion.
Thus, the correct answer is (B) decrease and increase .