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See Explanations....

Explanation:

NOTE => See attached graph.

1. 25-Turn Electromagnet

a. Strength of Batteries 1.5v vs 6.0v

6v:1.5v => 4:1 voltage ratio => That is, 6v battery is 4x voltage of 1.5v battery.

b. Ratio of Paperclips Picked Up

23clips(6v):6clips(1.5v) => 3.8clips(6v):1clip(1.5v)

That is, the 6 volt battery-EM picks up 3.8x more clips than the 1.5v battery-EM assuming constant number of coils given.

c. Empirical Relationship

From data given, an increase in voltage (V) produces a proportional increase in number of paperclips (C) picked up.  Such is a Direct Relationahip between #Clips Picked Up and Battery Voltage.

Thus, C ∝ V => C = kV => k = C/V

Therefore, for k₁ = k₂ => C₁/V₁ = C₂/V₂

2. 50-Turn Electromagnet

a. Ratio of turns between electromagnets.

50-Turn EM : 25-Turn EM => 2:1 ratio of turns between electromagnets

b. Ratio of Paperclips Picked UP

Between maximum voltage and minimum voltages for 50-Turn EM

=> 47 clips (6v) : 13 clips (1.5v) => 3.6:1 ratio...

That is, the 6v battery-EM, on the average, picks up 3.6x more clips than the 1.5v battery-EM given 50-Turn EM.

3. Doubling number of coils:

Doubling the number of coils, on the average, doubles the number of clips picked up by either electromagnet.

      Voltage              50-Turn EM      25-Turn EM     Pick Up Ratio

         1.5v                         13                       6                        2:1

         3.0v                        26                     12                        2:1

         4.5v                         31                    15.5                      2:1

         6.0v                         47                     23                      2:1  

4. Doubling Voltages:

Doubling voltage doubles the number of clips picked up by a given electromagnet regardless of the number of coils associated with magnet.

25-Turn EM          Voltage      Clips Pkd UP

                                1.5v                 6      

                                3.0v               12                

Double voltage => Doubles # clips picked up*

50-Turn EM          Voltage      Clips Pkd UP

                                1.5v                 13      

                                3.0v                26                

Double voltage => Doubles # clips picked up*

*Double Voltage => Double #clips picked up regardless of number of coil turns in electromagnet.

5.  Predicting number of clips for 7.5v battery-EM

Using linear trendline equations shown on attached graphs...

#clips 50-Turn EM = 7.1333V + 2.5 = 7.1333(7.5) + 2.5 ≅ 56 clips

#clips 25-Turn EM = 3.6333V + 0.5 = 3.6333(7.5) + 0.5 ≅ 27 clips

Overall => 2:1 ratio of clips between 50-Turn EM and 25-Turn EM

6. Slope values based upon trendline equations shown on attached graphs are ...

25 Turn EM => slope = 3.6333

50 Turn EM => slope = 7.1333

Approximately a 2x increase upon doubling the number of coils of electromagnet.

Predicting number of clips for 1V electromagnets:

#Clips (1volt) 25-Turn EM = 3.6333(1) + 0.5 = 3.6833 ≅ 4 clips

#Clips (1volt) 50-Turn EM = 7.1333(1) + 2.5 = 9.6333 ≅ 10 clips

Shows to be consistent with double #clips with double number of coil turns.

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